Let me start with a word: Alief. An Alief is like a belief, but instead of it being something you think is true it's something you feel is true (it's a pun you see - a-liefs come before b-liefs).
Alief is a very good word that I use a lot. If you don't have it yet, you're welcome, because you'll see it everywhere now.
It's particularly interesting to notice where your aliefs and beliefs diverge. Sometimes you believe things but you don't alieve them (e.g. you might know you are good at your job but experience impostor syndrome anyway), sometimes you alieve things but you don't believe them (you might think your friends secretly hate you).
As you might have noticed from those examples, it is often my experience that our beliefs are more functional than our aliefs. This isn't always the case. For example you might get a weird vibe off someone and go "No I'm just being unfair" and try to ignore it, only to later discover that they are indeed a massive creep who you are better off ignoring. You alieved that they were dangerous but you didn't believe it, and the belief let you astray. Hunches and intuition in general are often like this - you might have no objective reason to believe that a particular direction is promising, but you've got a strong hunch (an alief) that it would be.
An alief is just a felt sense (in the Gendlin sense) that something is true, but it's a felt sense that is usefully framed as being like a belief where you've just skipped over the surrounding propositional infrastructure.
One of the advantages of beliefs is that we can use metacognition to examine them. It's not actually true that we arrive at beliefs through simple propositional reasoning, but we're much more able to pretend that we are - we can put forth logical arguments for them, sometimes backfilling to construct ones where we realise we don't have them, etc. Often once we attack a belief with metacognition we might find it degrades to an alief - we no longer think it's true, but we feel it's true.
As this process suggests, there isn't actually a clear distinction between aliefs and beliefs: Beliefs are formed on aliefs, aliefs are informed by beliefs, and the whole thing is really a much more complicated network of interplay between felt sense, explicit verbal thought, and action.
It is perhaps better to think of alief and belief as directions - the more your relationship to a proposition is built on felt sense, the more like an alief it is, the more it is built on explicit reasoning, the more like a belief it is, but the reality is that there are no pure aliefs and no pure beliefs, only things that are more or less each (note that alief and belief are not opposite directions, but orthogonal ones).
Regardless of how pure they are though, we will still run into situations where we find that what we think is true is in conflict with what we feel is true. What do we do then?
A difficulty that people may have is that because they can explicitly work with beliefs through metacognition, they can update their beliefs relatively easily, but they have no such tools for updating their aliefs. Thus in a conflict one of two things will happen: Their beliefs will change, or they will be stuck with unhelpful aliefs (which they believe to be false but cannot shake).
This is where tools like Focusing come in, allowing us to work better with the felt sense.
In this sense, things such as the tools I outlined in Emotional reactions as legacy code can be thought of as the other half of the toolkit for ensuring coherence between your beliefs and your aliefs, bringing your aliefs into alignment with your beliefs when it is the feeling that is wrong and that needs to change.