I am interested in tuples of real or complex $d\times d$ matrices which are *irreducible* in the sense that their constituent matrices do not admit a common invariant linear subspace whose dimension is strictly between $0$ and $d$. In particular, I am interested in how the property of irreducibility is affected by some tensor-algebraic procedures. My question is:

If the $N$-tuples $(A_1,\ldots,A_N)$ and $(B_1,\ldots,B_N)$ are irreducible, when is the $N$-tuple $(A_1\otimes B_1,A_2 \otimes B_2,\ldots, A_N \otimes B_N)$ irreducible?

If we are working in the real context then it is trivially true that the tuple of pairwise tensor products is not necessarily irreducible: for example, if $N=1$, $A_1e_1=e_2$, $A_1e_2=-e_1$ then the $1$-tuple $(A_1)$ is irreducible, but $A_1 \otimes A_1$ preserves the span of $\{e_1\otimes e_2,e_2\otimes e_1\}$. Indeed, the $1$-tuple $(A_1\otimes A_1)$ is obviously reducible because every $4\times 4$ real matrix has a nontrivial invariant subspace.

So far I have not found an example in the complex case where the original tuples are irreducible but the product tuple is not. So: when does irreducibility pass to the pairwise tensor product if we are working over the complex field?