Classification of all $\mathcal{N}\ge 3$ moduli space orbifold geometries at rank 2

We classify orbifold geometries which can be interpreted as moduli spaces of four-dimensional $\mathcal{N}\ge 3$ superconformal field theories up to rank 2 (complex dimension 6). The large majority of the geometries we find correspond to moduli spaces of known theories or discretely gauged version of them. Remarkably, we find 6 geometries which are not realized by any known theory, of which 3 have an $\mathcal{N}=2$ Coulomb branch slice with a non-freely generated coordinate ring, suggesting the existence of new, exotic $\mathcal{N}=3$ theories.