Intermediate submodels which do not satisfy AC The following is known:
Theorem. Suppose $V[G]$ is a generic extension of $V$ by a set forcing, and let $N$ be a model of $ZFC$ with $V\subseteq N\subseteq V[G].$ Then $N$ is a generic extension of $V$ by a set forcing, in particular $N=V[A],$ for some set of ordinals.
It seems that the above theorem is not true if $N$ does not satisfy $AC$. In fact the following abstract is given in a talk by James Cummings (see http://settheory.mathtalks.org/cmu-math-logic-seminar-tues-11-september/):
If $c$ is Cohen-generic over $L$,  then  there is a transitive class model $M$ of $ZF$ intermediate between $L$ and $L[c]$ which is not of the form $L(A)$ for any $A.$ 
Does anyone know a proof of this fact?
 A: To my knowledge there is no written proof of this fact. I have all the available notes, which include a very very scattered description of $V_{\omega+1}$ and $V_{\omega+2}$ of this model $M$, and a single lemma which is used to proceed through successor of singular cardinals.
I am working on rebuilding this model in a cleaner method, and I am relatively close to finishing (some technical constructions are needed to finish the outline, but the idea itself is completely finished).
With luck I might actually finish this soon, and I could write a reasonable outline announcement (and then a full detailed accounts of the construction).

I should probably add that I asked all the people involved in the construction of this model, The Bristol model, and what I was told by everyone of them is that it started as some general idea to play with, and by the time they realized someone should be writing things down they already did a lot of the work, so it didn't survive into the notes.

Update (April 25, 2017):
It's on arXiv now.
