If the following is true, it is probably well-known to the experts. Nevertheless, I could not find a reference for it.

Suppose $P$ and $Q$ are $A_{\infty}$-operads in topological spaces and $X$ is a $P$-algebra as well as a $Q$-algebra. Moreover, assume that the actions of $P$ and $Q$ on $X$ commute, i.e. the following diagram is *strictly* commutative:

$$ \begin{gather} Q(n) \times P(m) \times X^{nm} & \to & Q(n) \times X^n \\\\ \downarrow & & \downarrow \\\\ P(m) \times X^m & \to & X \end{gather} $$

where the upper horizontal arrow uses the "diagonal action" of $P$ on $X^n$ and likewise for the left vertical arrow. Now I can deloop with respect to the operad $P$ to obtain $B_PX$, similarly I can obtain $B_QX$, where I either use the delooping machine of Boardman and Vogt or the bar construction - your choice.

Is it true that $B_PX$ carries an action of the operad $Q$ in this case?

Somehow this question boils down to the problem, how well products are respected by the common delooping machines. For example, I know that $B(G_1 \times G_2)$ is *homeomorphic* to $BG_1 \times BG_2$ for groups (or monoids). Is this still true, if $G_i$ are just $A_{\infty}$-spaces?