I would like to know how to obtain not well-founded $\omega$-models of ZFC.
Are there any books about it? Other references to the literature are also welcome.
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Once you know that the consistency of "There is a transitive model of ZFC" implies the consistency of "There is an $\omega$-model of ZFC", assuming the existence of the former will provide you with the existence of the latter.
Of course, it is consistent that there are no $\omega$-models, even if ZFC is consistent. So just obtaining them out of a model of ZFC is impossible.
But here is a nice way to get what you want. Start with a transitive model of ZFC, $M$. Fix some regular $\kappa>\omega$, and now add a generic ultrafilter and consider the generic ultrapower of $M$. It will have critical point $\kappa$, so it remains an $\omega$-model, but it will be ill-founded, as long as you didn't use a precipitous ideal for your forcing.