Generics over countable transitive models in ZF

Set $D_n=e(n)$ if e(n) is a dense subset of P, and $D_n=P$ otherwise. Every ground-model dense subset occurs. For q in P let $h(n,q)=e(k)$ for the least k such that $e(k)\in D_n$ and $e(k)\le q$. Transitivity gives $P\subseteq M$, so surjectivity of e and density ensure such k exists. This is a uniquely defined rule, not a choice function obtained by AC. Recursively put $p_0=p$ and $p_{n+1}=h(n,p_n)$.

Put $G=\{q\in P:\mathrm{\exists }n{p}_n\le q\}$. Then p belongs to G; transitivity gives upward closure; and the term with index max(n,m) strengthens any two members witnessed by n,m. Thus G is a filter. Each $p_{n+1}\in D_n\cap G$, so F1 makes G M-generic. Neither the sequence nor G is asserted to be an element of M.