I think that the following statement is true, but I do not know how to prove it.

Let $\mathfrak{g}_1$ and $\mathfrak{g}_2$ be two real simple Lie algebras. If $M$ is a (infinite dimensional) complex simple $\mathfrak{g}_1\oplus\mathfrak{g}_2$-module, then $M\cong M_1\otimes M_2$ where $M_1$ and $M_2$ are simple modules of $\mathfrak{g}_1$ and $\mathfrak{g_2}$ respectively.

This ought to be a well known result if it is true. As is known to all, it is true for finite groups and compact groups.