Forcing is a relative model construction method for models of $ZF$ as a particular first order theory using models of another first order theory (forcing companion) that in this case is the theory of partial orders ($PO$).

Is it possible to develope forcing for other first order theories? In fact I am asking for possible relative model constructions of a first order theory without finite models like $T$ (instead of $ZF$) using models of a "forcing companion" theory like $S$ (instead of $PO$). This model construction method produces countable models of $T$ with special properties using a given model of $T$ and a given model of $S$.

Obviously if we define a uniform "forcing companion theory" $S$ for an arbitrary first order theory $T$ then in many cases when countable models of $T$ are not too board up to isomorphism the "forcing theory of $T$" is just a trivial method and produces no interesting models of $T$. But in this question my focus is on first order theories with wild countable and countable models like $ZF$ and $PA$.

Every references in this frame work is welcome.