Given a family of forcing notions $(P_i)_{iin I}$ we can take the product $P:=prod_{iin I}P_i$ as a forcing notion to create a generic filter of the form $G=(G_i)_{iin I}$ such that for each $iin I$ the projection $G_i$ corresponds to the generic filter created when forcing with $P_i$. This is called product forcing and allows us to adjoin several different types of generic objects at once. (For a more detailed discussion of the subject see Product forcing and generic objects)

Now my question is if and how product forcing can be combined with symmetric forcing. Assume we have a family of forcing notions as above and a family of groups $(mathcal{G}_i)_{iin I}$ as well as $(mathcal{F}_i)_{iin I}$ such that $mathcal{G}_i$ is a subgroup of $Aut(P_i)$ and $mathcal{F}_i$ is a normal filter on $mathcal{G}_i$ for all $iin I$. Can we just define $P$ as above with $mathcal{G}:=prod_{iin I}mathcal{G}_i$ acting on $P$ componentwise and $mathcal{F}simeqprod_{iin I}mathcal{F}_i$ as a normal filter on $mathcal{G}$ ?

E.g. consider Cohen’s original symmetric model of $ZF+neg AC$ where he adjoins countably many generic reals and then proceeds to construct an infinite subset $Asubset mathbb{R}$ without any countably infinite subsets. Then the construction described above should allow us to adjoin $I$ many such sets $(A_i)_{iin I}$ at once.

Are there any complications one might encounter of with this type construction (i.e. symmetric product forcing)? Is there any literature on the subject?