  Mott-insulating transition metal oxides containing $t_{2g}^4$ ions with
strong spin-orbit coupling were recently demonstrated to display unusual
magnetism due to a dynamical mixing of the low-energy multiplet states via
exchange processes. Here we derive exchange interactions in the situation where
a tetragonal or trigonal crystal field selects a single relevant excited state
on top of the singlet ionic ground state, producing an effective
spin-$\frac12$. We show that these moments universally obey antiferromagnetic
transverse-field Ising model (TFIM) with an intrinsic transverse field
generated by the splitting of the two ionic singlets. Using Ru$^{4+}$ as a
example ion, we provide quantitative estimates of the exchange and illustrate
the emergent TFIM physics based on phase diagrams and excitation spectra
obtained for several 2D lattices - square, honeycomb, as well as for the
frustrated triangular lattice.
