Intermediate models with deep failure of choice

The following question was asked by Grigorieff: Suppose V is a ZFC model and $V[G]$ is a set-generic extension of V. Can there be a ZF model N so that V⊂ N ⊂ V[G] yet N is not equal to $V(A)$ for any set A∈ V[G]? The first such model was constructed by Karagila. This is the so-called Bristol model, an intermediate model between L and $L[c]$ 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 ⊂ V[c] so that N is not equal to $V(A)$ for any set A∈ V[c], 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 $V[c]$.