Are all supergravity theories Yang-Mills squared?

Using simple symmetry arguments we classify the ungauged $D=4$, $\mathcal{N}=2$ supergravity theories, coupled to both vector and hyper multiplets through homogeneous scalar manifolds, that can be built as the product of $\mathcal{N}=2$ and $\mathcal{N}=0$ matter-coupled Yang-Mills gauge theories. This includes all such supergravities with two isolated exceptions: pure supergravity and the $T^3$ model.