In which ways do anagamis still experience ignorance?

Clear understanding of the 4 Noble Truths (ignorance is not knowing the 4 Noble Truths.) or the lack of ignorance is only present in an Arahat. What I can say is that an Anagami's comprehension of the 4 Noble Truths is not complete in comparison to an arhat. To what extent the anagami lack ignorance in comparison to a sotapanna is difficult to say. The difference between an anagami and a sotapanna is the presence or lack of other factors. Ignorance is present in both.

Following are some abilities of the Buddha:

4. The Tathagatha Buddha has realized the way how worldly beings match with each other; that is the ignoble coalesce with the ignoble, the ungrateful ones with the ungrateful ones, the bad with the bad, the good with the good, and the various and different matching natures as they really are. This is His Fourth Power.

5. The Tathagatha Buddha has realized the diversity of beings and their various types of behaviors, desires, and emotions. It is in accordance with this knowledge that He teaches the Dhamma so that beings may best understand His Teachings. This is His Fifth Power.

So only the Buddha may know the true difference between the level of the ignorance factor between a sotapanna and anagami.

Householder gooiditnietweg, interested,

maybe it's good to make clear that one does don't really experiences ignorance or ignorance as ignorance (not knowing what should be known for release) but the effects of such: i.e. suffering.

Once one sees suffering clearly and understands it's arising, path and fading, it decays.

An Anagami (One no more after a household, new stand) has practiced the path to an extend where "only" this, in relation to previous states of Noble-hood, remains (even very refined), spoken in mental fetters:

• rūpa-rāga (desire after form)
• arūpa-rāga (desire after non-form)
• māna (pride)
• uddhacca (restlessness)
• avijjā (not knowing)

Aspects which are no more present:

• dosa (aversion)
• maccariya (stinginess, like all Noble Ones)

Falsehoods:

• micchā-sakappa (wrong resolves)
• pisunā-vācā (malicious speech)
• pharusā-vācā (pharusa vaca)

... see best Table of Mental Fetters, etc. Abandoned by Each Path-Knowledge, generously given by Bhante Ñāṇadassana Thero

Aside of possible old kamma arising that he might not suddenly enought abound by seeing, he still might cause himself suffering by actions in:

• samphap-palāpa (useless involvement in not conductive speech = burdensome talks)
• micchā-vāyāma (wrong effort)
• micchā-sati (wrong remembering)
• micchā-samādhi (wrong concentration)
• micchā-vimutti (wrong liberation)
• micchā-ñāṇa (wrong seeing)

Although one having joined the path of Anagami and no more destinated to return into the world of sensualtity, heading direct to final liberation latest in the next existence, as long as not reached Arahatship, path and fruits, he may still be touched by suffering, subject to it, but by far no more in ways which could be taken on actions which lead to lower existences then the arupa-existances.

Ill-will is something that do no more exist in an Anagami and he is free from perceiving such as enemies. Another may perceive mother or father, acting helpful for one as a child, as ill-will acting, but such is the nature of childs (worldlings). So in regard of this aspect of householders question, no, an Anagami does never act for others disadvantages but the opposite, even leaving already this sensual world and fights on form behind. He has no more interests in gain an hold anything what ordinary beings are after and could therefore not perceive such as an enemy for anyone, sees just blind fighting over dirt perceiving it as worthy.

Anāgāmī: the 'Non-Returner'

(1) “He may, immediately after appearing there (in the Pure Abodes) or without having gone beyond half of the life-time, attain the holy path for the overcoming of the higher fetters. Such a being is called 'one who reaches Nibbāna within the first half of the life' (antarā-parinibbāyī).

(2) “Or, whilst living beyond half of the lifetime, or at the moment of death, he attains the holy path for the overcoming of the higher fetters. Such a being is called 'one who reaches Nibbāna after crossing half the life-time' (upahacca-parinibbāyī).

(3) “Or, with exertion he attains the holy path for the overcoming of the higher fetters. Such a being is called 'one who reaches Nibbāna with exertion' (sasaṅkhāra-parinibbāyī).

(4) “Or, without exertion he attains the holy path for the overcoming of the higher fetters. Such a being is called 'one who reaches Nibbāna without exertion' (asaṅkhāra-parinibbāyī).

(5) “Or, after vanishing from the heaven of the Aviha-gods (see Suddhāvāsa), he appears in the heaven of the unworried (atappa) gods. After vanishing from there he appears in the heaven of the clearly-visible (Sudassa)-gods, from there in the heaven of the clear-visioned (Sudassī) gods, from there in the heaven of the highest (akaniṭṭha) gods. There he attains the holy path for the overcoming of the higher fetters.

Such a being is called 'one who passes up-stream to the highest gods' (uddhamsota-akaniṭṭha-gāmī).”

One may ask how the way of life of an Anagami, still bound by obligations in the world, life as non-Ascetic: well there are stories of a Devoted having reached this state as a householder but still had to look after his mother. Abstaining from wrong livelihood he used to collect what the river daily brought on the banks and could in this way also support his old and sick mother. An Anagami is a renouncer and would not be found under busy householder, traders and ordinary worker and actually lead a mode of an recluse, if possible very near the/a good following monastic Sangha, doing services and avoid involvements with those not devoted toward Nibbana.

As for monks and homeless, such a person would not be found under city or scholar monks but is one devoted to Dhutonga and remote and not organiced practice which should be not mistaken with the modern "Forest Tradition" which uses this label but is actually void Anagami but after becoming.

(Note: this is not given for exchange, stacks, trade or entertainment but as a means for liberation from this wheel.)

In simple terms, Anagami have seen that the body is suffering proper, and have released from worldly desire (body is how we experience the world). They have not yet seen the noble truth of suffering that consciousness is also suffering proper, and thus have not released from consiousness (vinnana). They still see wrongly that having samadhi is happy, not suffering.

The removal of fetters. A glimpse of insight. As such, the understanding of the Three Great Marks acts as the tool to get over the threshold of the first three fetters. Observing the Four Foundations of Mindfulness in light of (or with the understanding of) the Three Great Marks, he steps onto the path of Sotapanna. The understanding of the first three fetters as fetters has been realized, however it is not fully realized in all areas of his life. Similarly with the higher fetters, one may know the right thing to do it but karmic seeds remain that need to be worked off. The Anagami stil has feelings of desire and energy, however they are represented more as conceit, restlessness and ignorance, and subtle craving. The understanding of the first 3 fetters is clearer. The realization that strong desire and ill-will has departed. The Three Great Marks continue to assist, until perfect clarity of the Truth of annica, anatta and dhukka are fully realized; that is, impermanence, change, and attachment or aversion to anything that is subject to annica and anatta, will be dhukka. Although the understanding of this becomes understood at Sotoapanna, the understanding needs to be consistently reinforced. It is an understanding that gets stronger and clearer, until ultimately all ignorance is removed. Further, much like recognizing signs along the way, it depends on the strength of intuition of the individual. Some recognize a sign well before it is reached; some recognize it as it passed; and others recognize it after the change has come. Some foresee the arising, others recognize it in the moment, and others in reflection. Either way, it is understood.

The one sutta that came to mind regarding your request had neither avijja nor anagami, but I think it may have an answer to your question according to my reading of the sutta:

In S​N​22.89, we read about Venerable Khemaka, who is ill and dealing with the conceit, "I am".

For when it comes to the five grasping aggregates I’m not rid of the conceit ‘I am’. But I don’t regard anything as ‘I am this’.

Because of this, we might infer that Ven. Khemaka was a non-returner. Yet Khemaka understood the escape even though he was ignorant of its fruition:

Although a noble disciple has given up the five lower fetters, they still have a lingering residue of the conceit ‘I am’, the desire ‘I am’, and the underlying tendency ‘I am’ which has not been eradicated.

The subtlety of ignorance in non-returners is compared with the lingering of a cleansing scent:

Although that cloth is clean and bright, it still has a lingering scent of salt, lye, or cow dung that had not been eradicated.

Ignorance as the last fetter is merely a broad fetter including any type of attachment or self-view that still may arise.

For example, the life of the anagami may be threatened with deadly violence and the idea "I am" may arise in the anagami just before the anagami is beheaded with a sword.

Or the anagami may see many individual life forms slaughtered in war and may think those life forms or aggregates are "beings" or "people". For example, an anagami may believe 3,000 "beings" ("satta") "died" ("marana") on 9/11 where as an Arahant would perceive 15,000 aggregates (khandha) or 18,000 elements (dhatu) ending on 9/11.

There is a sutta (somewhere) where an anagami did not attain arahantship because they got excited about the Dhamma and had the urge to tell their "uncle" (other other relative). The anagami still viewed their "uncle" as "a being" or "person".

Or an anagami may get attached to the Dhamma, such as imputing "self-view" or "personhood" upon an individual that misrepresents the Dhamma doctrinally.

Ignorance is not-knowing the 4 noble truths and not knowing the three characteristics. The 1st noble truth is summarised as "attachment to the five aggregates". The three characteristics are impermanence, unsatisfactoriness & not-self. Therefore, if an anagami attaches to anything as "self", ignorance remains.

According to MN 9 (below), as long as you still have fermentation or effluents, you would still have ignorance, and vice versa (also see this question). Ignorance is a very deeply ingrained and deeply lingering fetter.

From the origination of fermentation comes the origination of ignorance. From the cessation of fermentation comes the cessation of ignorance. .....

From the origination of ignorance comes the origination of fermentation. From the cessation of ignorance comes the cessation of fermentation.

There is a vicious cycle between ignorance and fermentation.

There seems to be four types of fermentations in the Dhammawiki page of fermentation:

1. The mental fermentation of sense-desire (kāmāsava), Ex: 'All is pleasant'
2. The mental fermentation of desiring existence (bhavāsava), Ex: 'Being is good'
3. The mental fermentation of wrong views (ditthāsava), Ex: 'My opinion is best'
4. The mental fermentation of ignorance (avijjāsava). Ex: 'Suffering exists not'

According to this wikipedia page on ten fetters, the Anagami still has the fetters of lust for material existence, lust for immaterial existence, conceit, restlessness and ignorance.

So, the Anagami would have overcome the mental fermentation of sense-desire (kāmāsava) and the mental fermentation of wrong views (ditthāsava). However, he would still be challenged by the mental fermentation of desiring existence (bhavāsava). I take it that conceit is related to bhavāsava.

What does this mean?

The Anagami has overcome self-view, attachment to rituals, doubt, sensual desire and ill will. But the Anagami still has the mental fermentation of wanting to exist. And this will cause ignorance to continue.

You know the Four Noble Truths very well, dispelled wrong views (including self-view), overcame sensual desire and overcame ill will. But when someone threatens your life, you may get scared. Why? It's because you still want to exist.

I've read that one time Ven Anuruddha being a non-returner talked to Sariputta and it went something like this:

"With clairvoyance that is purified and surpasses the human, I survey the entire world-system. My energy is roused up and unflagging, my mindfulness is established and lucid, my body is tranquil and undisturbed, and my mind is immersed in samādhi. But my mind is not freed from the defilements by not grasping.”

Venerable Sariputta then answers:

“Well, Reverend Anuruddha, when you say: ‘With clairvoyance that is purified and surpasses the human, I survey the entire world-system,’ that’s your conceit. And when you say: ‘My energy is roused up and unflagging, my mindfulness is established and lucid, my body is tranquil and undisturbed, and my mind is immersed in samādhi,’ that’s your restlessness. And when you say: ‘But my mind is not freed from the defilements by not grasping,’ that’s your remorse. It would be good to give up these three things. Instead of focusing on them, apply your mind to the deathless.” - AN3.130 https://suttacentral.net/an3.130/pli/ms