Is it possible to take a forcing extension which is elementarily equivalent to the ground model? Here I'm assuming the extension is proper, that is, it adds a new set.
It's clear it can't be an elementary extension (the forcing notion has a generic filter in the extension but not the bottom), which is why I ask about equivalence. Some hypotheses would prevent this (no forcing extension of a model of V=L is still a model of V=L), so this is really a question about consistency.
It's also clear by a pigeonhole principle argument that since there are many forcing extensions but few complete extensions of ZFC, some pair of them must have the same theory. But this doesn't mean they have the same theory as the ground model.
Any thoughts?
Sure. Consider when $V=L[c]$ where $c$ is a Cohen real over $L$. If we force to add a second Cohen real $r$ over $V$, the result is again a Cohen extension of $L$.
And because the Cohen forcing is homogeneous, every two extensions are elementarily equivalent. So both $V$ and $V[r]$ are elementarily equivalent.