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 $M$?

The Bristol model: an abyss called a Cohen real. J. Math. Log.18(2018), no. 2, 1850008, 37 pp. (Take a look at section 7.2 for a brief comment on this matter.) $\endgroup$