Working in ZFC. Say $\mathbb V$ is the ground model, and $\mathbb V[G_1]$ and $\mathbb V[G_2]$ are forcing extensions. Is there a forcing extension $\mathbb V[H]$ containing $\mathbb V[G_1]$ and $\mathbb V[G_2]$?
I'm concerned about definitions of the form "there is a forcing extension where (condition) holds" where (condition) is something that is upwards (but not downwards) absolute. My context is model-theoretic but I don't think that's relevant.
I'm not a set theorist, so apologies if the answer is quite obvious. My first thought is just to say $H=G_1\times G_2$, but I don't know if that's necessarily a well-behaved idea?
The answer is no in general. For example, work over a countable transitive model $M$ for simplicity; then we can find two Cohen-over-$M$ reals, $G_0$ and $G_1$, whose join $G_0\oplus G_1$ computes a well-order of $\omega$ of order type the height of $M$. (For detail, see Andreas Blass' answer to this old question of mine: https://mathoverflow.net/questions/38648/a-question-about-iterated-forcing.)
More succinctly, just because $G_0$ and $G_1$ are $\mathbb{P}_0$ and $\mathbb{P}_1$-generic does not mean that $G_0\times G_1$ is $\mathbb{P_0}\times\mathbb{P_1}$-generic. (For a silly example of this, take $G_0=G_1$! Of course, the above example is much better.)
What is true is that - again, working over a countable transitive model for simplicity - given any generics $G_0$ and $G_1$ and any poset $\mathbb{P}$, there is a $\mathbb{P}$-generic $H$ which is mutually generic with $G_0$ and mutually generic with $G_1$ (basically, think "the intersection of two comeager sets is comeager"). For many applications, this suffices.
Indeed, often the following even weaker statement is enough. Suppose $\varphi$ is true in $V[G_0]$. Then there is some condition $p\in G_0$ which forces $\varphi$. Similarly, if $\varphi$ fails in $V[G_1]$, then there is some condition $q\in G_1$ forcing $\neg\varphi$. Now, $G_0$ and $G_1$ may not be mutually generic, but you don't care: force with the posets below the relevant conditions! That is, supposing $\mathbb{P}_0$ and $\mathbb{P}_1$ were the forcings corresponding to $G_0$ and $G_1$ respectively, let $$\mathbb{Q}=\{(r, s): r\in\mathbb{P}_0, s\in\mathbb{P}_1, r\le p, s\le q\}$$ ordered componentwise, and let $H=H_0\times H_1$ be $\mathbb{Q}$-generic over $V$. (As a side note, recall that $V[H_0\times H_1]=V[H_0][H_1]=V[H_1][H_0]$, so your upwards absoluteness should be able to come into play.)