How does one accomplish a reality not governed by the mind when thought is required to function?

You cannot, not at least until you are enlightened & choose to do so.

What you can do is cease identification with the thought; detach yourself from the one who is doing the thinking. Karma will still play out for you, things happen, situations occur, thoughts arise. You can't really stop this.

The waves still as a consequence of following the path, you cannot willfully curb thought.

How does one accomplish a reality not governed by the mind when thought is required to function?

There is no one who can govern. There are only realities which causing others.

So, the purification of understanding in realities is the only way to specify the using of "govern" word whether wrong or right using.

It's ok to say "governed by the mind" if the speaker realizing in every moment weather "there are only realities depending on other realities, nothing controllable itself".

It's wrong if the speaker don't realize even one moment, although the speaker avoid to say the word "govern".

And that ability required proficient Jhana and Abhidhammic detail. The proficient Jhana let the practitioner see ultimate realities in very advance, abhidhammic detail. The abhidhammic detail are seeing the smallest reality more than atom, fastest arising&vanishing more than the light speed, more than trillion time in a second.

It often seems that ultimate reality is only available in the absence of thought, yet interactions, work and daily functioning requires thought.

The ultimate reality is every where either thinking or not. The mind can know only a reality or a concept per arising, however it's conascence-wisdom can understand both of them at the same moment.

So, the thought is the opposite of only Jhana.

However, the unwholesome is the opposite of the insight-meditation as well, so it is important to do Jhana for pausing the thought. Jhana is easier than the insight-meditation, less detail, so it is good in short term to stop the unwholesome mind.

How does one curb thought when it is uncertain as to whether there is utility in the thought and that it is necessary to follow that thought to achieve something? Wouldn't this mean that ultimate reality can't be accomplished?

"Curbing thought" is Jhana. Jhana is required but not all the time.

It's important for insight meditation weather "how deep in detail that you can see the realities?".

But it's important for Jhana meditation weather "how long time that you can pause the thought?"

However, both Jhana-meditation and insight-meditation are important to do.

"governed by the mind" suggest that there is one who is governed. Once you enter the 2nd refuge of the 3 jewels of Buddhism (taking refuge in the Dharma) and the teacher and the teaching as localized sources melt away, you become one with all , therefore the continued teaching and realizations do not flow from a person or from a book anymore. The whole starts teaching herself.

2+2=4 not because your mind is localized and is able to calculate. The same way, things can be happening (urge to go to the bathroom) without the mind creating a story about it. Then "you aren't going to the bathroom", but rather "the whole is going to the bathroom". There is still "going to the bathroom", but a localized mind is not required to achieve the act.

Also IMPORTANT, none of these states are black/white, on/off, etc. In most cases there are partial realizations or degrees of achievements. Once you notice the gap (between what is and what is in the mind), you can widen that gap with practice and repetition. So the functionality of the mind can be slowly/quickly, gradually/at once be transferred/tasked to work as part of the whole, rather than work for the individual. The Eightfold Path gives you specific training to work on the mind and its functionality, to slowly erode its fascination with itself. Then it becomes a tool, rather than the seat of your god.