Does forcing equivalence imply isomorphic boolean completions?

Question

That two forcing notions with isomorphic boolean completions are forcing equivalent, ie. they yield the same class of forcing extensions, is well-known. But for some reason, I could not find much about the (possibly partial) converse. What I would like to know is this: if $\mathbb{P}$ and $\mathbb{Q}$ are forcing equivalent, antisymmetric partial orders, is it necessarily true that $RO(\mathbb{P}) \cong RO(\mathbb{Q})$ as boolean algebras? If not, would adding the condition that both $\mathbb{P}$ and $\mathbb{Q}$ are separative change the conclusion?