  We investigate tensor products of random matrices, and show that independence
of entries leads asymptotically to $\varepsilon$-free independence, a mixture
of classical and free independence studied by M{\l}otkowski and by Speicher and
Wysocza\'nski. The particular $\varepsilon$ arising is prescribed by the tensor
product structure chosen, and conversely, we show that with suitable choices an
arbitrary $\varepsilon$ may be realized in this way. As a result we obtain a
new proof that $\mathcal{R}^\omega$-embeddability is preserved under graph
products of von Neumann algebras, along with an explicit recipe for
constructing matrix models.
