TITLE: Forcing with Nontransitive Models
QUESTION [11 upvotes]: A common approach to forcing is to use countable transitive model $M \in V$ with $\mathbb{P} \in M$ and take a $G \in M$ (which always exists) to form a countable transitive model $M[G]$. Another approach takes $M$ to be countable such that $M \prec H_\theta$ for sufficiently large $\theta$ (and hence may not be transitive). For example, a definition of proper forcing considers such models. 
Forcing with transitive models are quite convenient since many absoluteness results can be used to transfer properties of $x \in M[G]$ which hold in $M[G]$ up to $V$. If $M \prec H_\theta$ is not transitive, then it is not clear what type of property that $M[G]$ can prove about $x$ transfer to $V$. For instance, if $M[G] \models x \in {}^\omega\omega$, is $x \in {}^\omega\omega$ in $V$? Of course, one remedy could be to Mostowski collapse everything and then use the familiar absoluteness for transitive models. For $x \in {}^\omega\omega$, one could use the fact that $M \prec H_\theta$ implies $\omega \subseteq M$ and hence the Mostowski collapse of $M[G]$ would maps each real to itself and then use absoluteness to prove that $V \models x \in {}^\omega\omega$ as well. Is there a more direct way to prove these type of result rather than collapsing the forcing extension, which seem to suggest one should have started by collapse $M$ before starting the forcing construction. 
So my questions are
1 First, if one chooses to work with countable $M \prec H_\theta$ are there any changes that need to made to the forcing construction and the forcing theorem as they appear in Kunen or Jech? Of course, the definition of a generic filter should be changed to meeting those dense sets that appear in $M$. 
2 I am aware that if $G$ has master conditions, then $M[G] \prec H_\theta[G]$? Is $H_\theta[G]$ just the forcing construction applied to $H_\theta$? As $G$ is not necessarily generic over $H_\theta$, it is not clear to me that the forcing theorem need to apply to $H_\theta[G]$ (or a priori $H_\Theta[G]$ models any particular amount of $\text{ZF}- \text{P}$, but since $M[G] \prec H_\theta[G]$, actually $H_\Theta[G]$ would model as much as $M[G]$.) In general without addition assumption like master conditions, does the relation $M[G] \prec H_\Theta[G]$ still hold. 
Also perhaps I am misunderstanding something, but since $\mathbb{P} \in M$, it appears that if $\theta$ is large enough, every $G \subseteq \mathbb{P}$ which is $\mathbb{P}$-generic over $M$ is already in $H_\Theta$. Would this not imply that $H_\theta[G] = H_\theta$ and hence $M[G] \prec H_\Theta$. Since $M \prec H_\theta$, $M$ and $M[G]$ models the exact same sentences. This surely can not happen.
Thanks for any help and clarification that can be provided.

REPLY [4 votes]: To summarize Joel's comments. 
Suppose that $M$ is a possibly nontransitive model of a sufficient fragment of ZFC like Zermelo with Choice and some Replacement. (Needs it to be an elementary submodel of some $H_\kappa$?) Let $P\in M$ be a forcing notion. In parallel to the usual inductive interpretation 
$$
t[G]=\{s[G]:\exists p\in G(\langle s,p\rangle\in t)\}
$$
and $M[G]=\{t[G]:t\in M\}$, we define the modified interpretation 
$$
t^M[G]=\{s^M[G]:\exists p\in G(\langle s,p\rangle\in t\cap M)\}
$$
and $M[G]^{\text{mod}}=\{t^M[G]:t\in M\}$. 
Note that $M[G]^{\text{mod}}$ is a transitive set and not necessarily an extension of $M$.
A filter $G\subseteq P$ is strongly $P$-generic over $M$ iff $G\cap D\cap M\ne\emptyset$ whenever a set $D\in M$, $D\subseteq P$ is dense in $P$.  Note: $G$ is not necessarily an ultrafilter in $P$ in this case!
Let $\pi=\pi_M:M\to M'$ be the Mostowski collapse, $\pi(x)=\{\pi(y):y\in x\cap M\}$, so $M'$ is a transitive set $\in$-isomorphic to $M$ by $\pi$. 
Let $P'=\pi(P)=\{\pi(p):p\in P\cap M\}$, a forcing in $M'$.
Theorem.
In these assumptions, let $G\subseteq P$ be strongly $P$-generic over $M$. Then 
(1) the set $G'=\{\pi(p):p\in G\cap M\}$ is $P'$-generic over $M'$ in the usual sense, and 
(2) $M'[G']=M[G]^{\text{mod}}$, in fact $t_M[G]=\pi(t)[G']$ for each $t\in M$.
Moreover if $A(x_1,\dots,x_n)$ is any formula and $t_1,\dots,t_n\in M$ then $A(t_1^M[G],\dots,t_n^M[G])$ is true in $M[G]^{\text{mod}}$ iff there is a condition $p\in G\cap M$ which $P$-forces $A(t_1,\dots,t_n)$ in $M$. 
$\Box$
TODO list. 

Prove the theorem carefully.
Study the relations between $M[G]$ and $M[G]^{\text{mod}}$. Still $M[G]$ is of a separate interest, albeit it is not only nontransitive but also not necessarily even an extensional model. (See an example in one of my comments.)
Possible applications. For instance in the case $M\prec H_\kappa$, $M$ countable, one can consider sets $G\subseteq P$ in $H_\kappa$, strongly $P$-generic over $M$ (but not over $H_\kappa$, of course) as in my arXiv paper on countable OD sets in Cohen's extensions.