Here's an example using random and Cohen forcing, denoted respectively by $\mathcal R$ and $\mathcal C$.  Consider the two-step iteration $\mathcal C * \dot{\mathcal R}$.  If $(c,r)$ are generic reals for this iteration, then $r$ is random over the ground model $V$.  Therefore the random forcing as constructed in $V$ completely embeds into $\mathcal C * \dot{\mathcal R}$.  The map takes the following form.  If $x$ is a code for a positive measure Borel set, and $A_x$ denotes the set coded, then we map $e : A_x \mapsto (1,\dot{A_{\check x}})$.  An important point is that while the generic Cohen real changes the interpretation of $A_x$, $c$ does not exclude any such $\dot{A_{\check x}}^c$ from being in $\dot{\mathcal R}^c$, since the statement that $x$ codes a set of positive measure is absolute.

So to show this is a counterexample, we just need to show that the ground-model measure algebra $\mathcal R_0$ (given in terms of Borel codes), is not a regular subalgebra of the random forcing in $V[c]$.  Suppose it were, then:

(1) The real $s$ given by $e^{-1}$ applied to the generic filter determined by $(c,r)$ would be random over $V[c]$.  This is because a regular embedding of a partial order $\mathbb P$ into a complete boolean algebra $B$ extends uniquely to a complete embedding of $\mathcal B(\mathbb P)$ into $B$, and any complete subalgebra of a measure algebra is a measure algebra.

(2) Forcing with $\mathcal R$ over $V[c]$ is equivalent to forcing with $\mathcal R_0 * \dot{\mathbb Q}$, where $\mathbb Q$ is some further (possibly trivial) forcing.  Thus $(c,s)$ is (Cohen $\times$ Random)-generic over $V$, and so $c$ is Cohen-generic over $V[s]$.

By a well-known argument, $c$ constructs a Borel code $y$ for a measure zero set that covers all ground model reals.  Thus in $V[s][c]$, $s \in A_y$, and therefore there is a code $\neg y \in V[c]$ for a measure one set such that $s \notin A_{\neg y}$, meaning $s$ is not random over $V[c]$.

I also have an example involing Suslin trees and collapsing $\omega_1$, but it seems more interesting if we can stick to random and Cohen.