I'm not sure I understand. How do you do it?
In my thought process, I quite literally stop myself, and say "ok, think about what you just said" to check myself. I literally initiate that loop. If I don't, then I'm not using my own mental agency, and just using my firm coded priors.
I will say that I do seem to have a stop, what you said is wrong logic check voice that pops up without me initiating it. But, it's unreliable, and not too much different than all the content monitoring system used for the streaming clients, that will terminate with "content violation" immediately after the "incorrect" words are sent. I don't think integration is important, just the behavior of the overall system.
In fact, in the large majority of the time, you don't process "as a loop" at all, you just continuously progress in your reflection without needing a "second voice" to retrigger you. The fact that sometimes we do this is just something we can do, not the result of something needed for our brain to work. For AI systems, this is something needed because the "answering" part is not able to do the loop on its own. And building a bigger system that combine an "answering" part and a "loop" part does not fit this, does not create a self-reflective system, it just makes a non-self-reflective system and a workaround bundled together.
It's a bit like if the "answering" part was unable to provide only one answer and was always producing plenty of different possible answers, including contradictory ones. Then, you can add an external part that will just pick one answer (and add it to the context so the next large set of answer will not be inconsistent), without any intelligence to it. The whole system will look like a human. But we know that the system is not "living" and "aware", because a "living" or "aware" system has its own opinion, while this system is just generating convincing sentence without seeing any hierarchy or value or meaning in each one.