  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.
