Intermediate models with deep failure of choice

The following question was asked by Grigorieff: Suppose V is a ZFC model and VG is a set-generic extension of V. Can there be a ZF model N so that V N VG yet N is not equal to V (A) for any set A VG? The first such model was constructed by Karagila. This is the so-called Bristol model, an intermediate model between L and Lc where c is a Cohen-generic real over L. Karagila further proves that the Kinna-Wager degree is unbounded in this model. We prove that such an intermediate extension can be found in a Cohen-generic extension of any ground model, fully resolving Grigorieff's question. That is, let V be any ZF model and c a Cohen-generic real over V. We prove that there is an intermediate ZF-model V N Vc so that N is not equal to V (A) for any set A Vc, the Kinna-Wagner degree of N is unbounded and, in particular, no set forcing in N forces the axiom of choice. Therefore, there are class many different intermediate models of ZF between V and Vc.