In "The Modal Logic of Forcing", Joel David Hamkins and Benedikt Löwe show that the ZFC-provable forcing principles are exactly those of the modal logic S4.2 (interpreting $\Diamond \phi$ as asserting that $\phi$ is forceable).
Do the $\mathsf{ZF}$-provable principles of forcing obey a different modal logic?
The definition of forcing is the same with and without the axiom of choice. And the truth lemma holds with and without the axiom of choice. Namely,
$$p\Vdash_\Bbb P\varphi\iff\text{For every }V\text{-generic } G\subseteq\Bbb P\text{ with }p\in G: V[G]\models\varphi$$
You can also consider iterations without choice, at least with a two-step iteration this holds absolutely no difficulties (compared to all sort of non-finite supports and so on). So the proof that $\sf S4.2$ is a subset of the modal logic of forcing in $\sf ZF$ is immediate.
In the other direction, clearly $\sf ZF$ cannot prove more than $\sf ZFC$. So you get $\sf S4.2$ immediately as a result.