TITLE: Models of ZF intermediate between a model of ZFC and a generic extension
QUESTION [10 upvotes]: Let $M$ be a countable model of $ZFC$ and $M[G]$ be a (set) generic extension of $M$. Suppose $N$ is a countable model of $ZF$ with 
$$M\subseteq N \subseteq M[G]$$
and that $N=M(x)$ for some $x\in N$; i.e., it is the smallest inner model of $M[G]$ which contains $x$ and $M$.
Is $N$ a symmetric extension of $M$?

REPLY [10 votes]: Yes, if $N=M(x)$ (taking the modern notation over Grigorieff's $M[x]$), then it is a symmetric extension. This is a very recent result of Toshimichi Usuba (see this and that).
However, by the Bristol model construction, this symmetric extension need not be obtained by the forcing that was used to get $M[G]$.