**Definition 1:** A class $\mathcal{K}$ of countable transitive models of $\text{ZF}$ has strong "joint forcing extension property" (JFEP) iff for all $M,N\in \mathcal{K}$ there are forcing notions $\langle\mathbb{P}, <\_{\mathbb{P}}\rangle\in M,\langle\mathbb{Q}, <\_{\mathbb{Q}}\rangle\in N$ and $\mathbb{P}$-generic filter $G$ on $M$ and $\mathbb{Q}$-generic filter $H$ on $N$ such that $M[G]=N[H]$. 


Also we say $\mathcal{K}$ has "weak JFEP" if $M[G]\equiv N[H]$.


**Definition 2:** A expansion $T$ of $\text{ZF}$ has strong/weak JFEP if the collection of its countable transitive models has strong/weak JFEP.


**Question:** Does $\text{ZF}$ have strong/weak JFEP? If not, is there any expansion of $\text{ZF}$ with strong/weak JFEP? In the other words, can one add strong/weak JFEP to $\text{ZF}$ by adding some new axioms like large cardianl assumptions or axiom of constructibility? 



Joint Forcing Extension Property
First, I note that the strong JFEP is the same as the medium JFEP, since two transitive sets such as $M[G]$ and $N[H]$ are isomorphic if and only if they are equal. 


Next, note that the strong property is just too much. 


**Theorem.** No consistent extension of ZF, which allows a model to be a forcing extension by adding a Cohen real, has the strong JFEP or the medium JFEP. 


Proof. For any countable transitive model $M$ of ZF, there are $M$-generic Cohen reals $c$ and $d$ such that $M[c]$ and $M[d]$ have no common forcing extension. One can build $c$ and $d$ so that each of them separately is $M$-generic, but the combination $c\oplus d$ codes a real that collapses all of $M$. This is explained in my answer to [this question](https://mathoverflow.net/questions/143413/is-there-a-forcing-closure). QED


This argument can be generalized beyond the forcing to add a Cohen real to any forcing that allows that coding argument to go through, and I believe this might include all forcing.


**Theorem.** There is a theory $T$ extending ZFC with the strong JFEP.


Proof. Assume without loss that there are transitive models of ZFC. Thus, there is a minimal transitive model. Let $T$ be the complete theory of the minimal transitive model $L\_\alpha\models ZFC$. Now, the point is that if $M$ and $N$ are models of $T$, then actually $M=N$, since both must have the form $L\_\beta$ for some $\beta$ but it cannot be that $\beta\gt\alpha$ and so both are equal to the minimal transitive model itself. Since $M=N$, of course it follows that they have a common forcing extension. QED


This idea seems to generalize to many other theories.


Meanwhile, it is easy to get the weak JFEP.


**Theorem.** Every complete theory trivially has the weak JFEP. 


Proof. If $T$ is complete and $M,N\models T$, then they are already elementarily equivalent and they are forcing extensions of themselves by trivial forcing. QED


