Suppose $A$ and $B$ are complete subalgebras of a complete boolean algebra $C$. Let $G \subseteq A$ be generic. In the extension $V[G]$, we can define the quotient algebras $B/G$ and $C/G$ in the usual way. Is $B/G$ a regular subalgebra of $C/G$?

The answer is yes in certain cases, such as if $A \subseteq B$ or $B \subseteq A$, or if $C$ is the completion of product of three partial orders, and $A$ and $B$ correspond to two of them.

I know a counterexample under MM, and I thought I had one in ZFC but I've lost it. Can you construct a counterexample?