What motivates a fully awakened person in a loving relationship?

It is impossible for a monk whose mental fermentations are ended to engage in sexual intercourse. (AN 9.7)

I think even an anagami would stop being interested in relationships because sensual desire is one of the fetter that is removed at non return. So an anangami and arahant are likely to leave the house and become monks. If the anagami has a wife or a husband before attaining non return then they will probably live as sister or brother and not as husband and wife.

Milindapanha III.19

"You say that if a layman attains arahantship he must either enter the Order that very day or die and attainparinibbàna. Yet if he is unable to find a robe and bowl and preceptor then that exalted condition of arahantship is a waste, for destruction of life is involved in it."

"The fault does not lie with arahantship but with the state of a layman, because it is too weak to support arahantship. Just as, O king, although food protects the life of beings it will take away the life of one whose digestion is weak; so too, if a layman attains arahantship he must, because of the weakness of that condition, enter the Order that very day or die."

So an arahant wouldn’t be able to live as a layman for long. I don’t remember which sutta but in that sutta the Buddha or someone said that arahant can only stay as layman for 7 days or they must enter Parinibanna.

Love is sensual desire and an anagami and arahant would completely be free of sensual desire so they won’t have a reason to stay in a relationship with his or her girlfriend. So love relationship is not possible for anagamis and arahants

I assume it would be karuna, compassion.

There's a Zen story, Is that so? -- probably not that "Zen master Hakuin" is counted an arahant, but I guess the motive displayed in that story is compassion.

Anyway I think we're told that the Buddha's own motive, per MN 26:

Then, understanding Brahmā’s invitation, I surveyed the world with the eye of a Buddha, because of my compassion for sentient beings.
Atha kho ahaṃ, bhikkhave, brahmuno ca ajjhesanaṃ viditvā sattesu ca kāruññataṃ paṭicca buddhacakkhunā lokaṃ volokesiṃ.

The motive presumably or by definition wouldn't be "sensuality", however.

And it's Theravada orthodoxy that any lay arahant must join the Order immediately.

Even if we assume no craving for sensuality, I'm not sure what you mean by "mutually intimate" in the question -- that phrase suggests to me some self-view (e.g. "I love you and you love me") -- perhaps these are part of a "thicket of views", which no longer bind an enlightened being.

Remark: This is not (yet?) a full answer, but more an extended comment.

For one impression of how might a loving relation be revered see "A.IV. 55 Nakulapitá und Nakulamátá" ; although the couple is not called arahants they are mentioned as "top on lay disciples" (see "A I.24") "The foremost of my laymen ...(...) ... who are intimate is the householder Nakula’s father.”
So there's no right-out disregard of being a loving couple.

Next it might be interesting a story about the (later) arahant (Maha) Kassapa about the problem of being married and having a desire for ascetic life at the same time. Hellmuth Hecker has compiled a story about this which I found some time ago at the access-to-insight site (I think) . Here is a part of that text on the biographic tale about Kassapa:

(...) Like the two chief disciples, Sariputta and Maha Moggallana, Maha Kassapa too descended from the brahmin caste, and again like them, he was older than the Buddha. He was born in the Magadha country, in the village Mahatittha, as the son of the brahmin Kapila and his wife Su-manadevi.

(This account of Maha Kassapa's early life is taken from the commentary to the Samyutta Nikaya.)

He was called Pipphali. His father owned sixteen villages over which he ruled like a little king, so Pipphali grew up in the midst of wealth and luxury. Yet already in his young years there was in him the wish to leave the worldly life behind, and hence he did not want to marry. When his parents repeatedly urged him to take a wife, he told them that he would look after them as long as they live, but that after their deaths he wanted to become an ascetic. Yet they insisted again and again that he take a wife, so to comfort his mother he finally agreed to marry -- on the condition that a girl could be found who conformed to his idea of perfection. For that purpose he shaped a golden statue of a beautiful woman, had it bedecked with fine garments and orna-ments, and showed it to his parents, saying: "If you can find a woman like this for me, I shall remain in the home life."

His parents approached eight brahmins, showered them with rich gifts, and asked them to take the image with them and travel around in search of a human likeness of it. The brahmins thought: "Let us first go to the Madda country, which is, as it were, a gold mine of beautiful women." There they found at Sagala a girl whose beauty equaled that of the image. She was Bhadda Kapilani, a wealthy brahmin's daughter, aged sixteen, four years younger than Pipphali Kassapa.

Her parents agreed to the marriage proposal, and the brahmins returned to tell of their success.

Yet Bhadda Kapilani also did not wish to marry, as it was her wish, too, to live a religious life as a female ascetic. Such identity between her aspiration and Pipphali Kassapa's may well point to a kammic bond and affinity between them in the past, maturing in their present life and leading to a decisive meeting between them and a still more decisive separation later on.

When Pipphali heard that what he had thought most unlikely had actually occurred, he was -- unhappy and sent the following letter to the girl: "Bhadda, please marry someone else of equal status and live a happy home life with him. As for myself, I shall become an ascetic. Please do not have regrets."
Bhadda Kapilani, like-minded as she was, independently sent him a similar letter.
But their parents, suspecting such an exchange would take place, had both letters intercepted on the way and replaced by letters of welcome. So Bhadda was taken to Magadha and the young couple were married. However, in accordance with their ascetic yearning, both agreed to maintain a life of celibacy. To give expression to their resolve, they would lay a garland of flowers between them before they went to bed, determined not to yield to sensual desire. This young wealthy couple lived thus happily and in comfort for many years. As long as Pipphali's parents lived, they did not even have to look after the estate's farms. But when his parents died, they took charge of the large property.
(...)

Now, further after the death of Pipphali's parents, Pipphali and also Baddha took the yellow robe and left home, but walked together. Then this happened:

(...) When walking on, Kassapa went ahead while Bhadda followed behind him. Considering this, Kassapa thought: "Now, this Bhadda Kapilani follows me close behind, and she is a woman of great beauty. Some people - could easily think, 'Though they are ascetics, they still cannot live without each other! It is unseemly what they are doing.' If they spoil their minds by such wrong thoughts or even spread false rumors, they will cause harm to themselves." So he thought it better that they separate. When they reached a crossroads Kassapa said: "Bhadda, you take one of these roads, and I shall go the other way." She said: "It is true, for ascetics a woman is an obstacle. People might think and speak badly about us. So please go your own way, and we shall now part."
She then respectfully circumambulated him thrice, saluted him at his feet, and with folded hands she spoke: "Our close companionship and friendship that had lasted for an unfathomable past comes to an end today. Please take the path to the right and I shall take the other road." Thus they parted and went their individual ways, seeking the high goal of Arahatship, final deliverance from suffering.
It is said that the earth, shaken by the power of their virtue, quaked and trembled.

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From: MAHA KASSAPA: FATHER OF THE SANGHA by Hellmuth Hecker Revised and enlarged translation from the German by Nyanaponika Thera

Translated and adapted from //Wissen und Wandel// XXI, 6 (1975) Wheel Publication No. 345 Copyright 1987, 1995 Buddhist Publication Society BUDDHIST PUBLICATION SOCIETY KANDY Sri Lanka

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So here we find something like "platonic love" between ascetics but in this instance much deeper, Hecker proposes the expression "karmic bond" for something which we might assume as an exotic example for a very deep arrangement of love over an already long chain of rebirthes...