Given a family of forcing notions $(P_i)_{i\in I}$ we can take the product $P:=\prod_{i\in I}P_i$ as a forcing notion to create a generic filter of the form $G=(G_i)_{i\in I}$ such that for each $i\in I$ 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 $(\mathcal{G}_i)_{i\in I}$ as well as $(\mathcal{F}_i)_{i\in I}$ 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 $i\in I$. Can we just define $P$ as above with $\mathcal{G}:=\prod_{i\in I}\mathcal{G}_i$ acting on $P$ componentwise and $\mathcal{F}\simeq\prod_{i\in I}\mathcal{F}_i$ 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 $A\subset \mathbb{R}$ without any countably infinite subsets. Then the construction described above should allow us to adjoin $I$ many such sets $(A_i)_{i\in I}$ 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?
Yes, there's a lot of this in the literature. Although very little in the ways of "abstract framework". This is something that was done essentially from the very early days of forcing, and you can find evidence of that in early papers.
In my works
You can find a more general treatment. Products are a particular case of an iteration, and the first paper deals with the case where the support is finite. In the case of a product, however, we can dispense with some of the difficulties in generalising iterations to arbitrary supports, and some of the work is done in the second paper.
In addition to that you can see products defined "by hand" in many places, it's easy to see that the definitions hold for any kind of symmetric systems (but the products are normally used with Cohen-style forcings). Here are some recent examples, mainly from my work that revolved this topic quite often, and older examples.
Hayut, Yair; Karagila, Asaf, Spectra of uniformity., Commentat. Math. Univ. Carol. 60, No. 2, 287-300 (2019). ZBL07144894.
Karagila, Asaf, Embedding orders into the cardinals with (\mathsf {DC}_{\kappa} ), Fundam. Math. 226, No. 2, 143-156 (2014). ZBL1341.03068.
Karagila, A., Fodor’s lemma can fail everywhere, Acta Math. Hung. 154, No. 1, 231-242 (2018). ZBL1413.03012.
Monro, G. P., Independence results concerning Dedekind-finite sets, J. Aust. Math. Soc., Ser. A 19, 35-46 (1975). ZBL0298.02066.
Roguski, Stanisław, A proper class of pairwise incomparable cardinals, Colloq. Math. 58, No. 2, 163-166 (1990). ZBL0706.03038.
Between all of these you'll see finite supports, countable (or $\kappa$-)supports, Easton supports, and you'll see that leaping towards anything else (which is now just other-kind-of-mixed support is really just the same).
In fact, we even have more power now since we can talk about changing the support in the product of the filters and the groups. You'd think that this means that we can say a whole lot more, but in fact, it's usually irrelevant.
In my paper about iterations I described a concept called "tenacity". Towards the end of my Ph.D. in one of the many discussions I had with Yair Hayut we decided to try and figure what really lies beneath that concept. And it turned out that every symmetric system is equivalent to a tenacious one. And that means that playing with different supports (i.e. finite support on the filters while using Easton on the forcing) is usually just equivalent to whatever smallest support you're using. Not necessarily always, but usually.
As for the Cohen model, that's a bit tricky. Each generic is a real, and we not only care about those, we also care about the set of all generics. So this is actually not a product, but rather an iteration of adding each real, violating choice by not adding the set of all reals, and then forcing to add the set of generics without its well-ordering. All this makes the approach of just thinking about it as a single extension a lot simpler.