  Excitonic magnetism involving superpositions of singlet and triplet states is
expected to arise for two holes in strongly correlated and spin-orbit coupled
$t_{2g}$ orbitals. However, uncontested material examples for its realization
are rare. Applying the Variational Cluster Approach to the square lattice, we
find conventional spin antiferromagnetism combined with orbital order at weak
and excitonic order at strong spin-orbit coupling. We address the specific
example of Ca$_2$RuO$_4$ using ab-initio modeling and conclude it to realize
excitonic magnetism despite its pronounced orbital polarization.
