  Hybrid Ising superconductor-ferromagnetic insulator heterostructures provide
a unique opportunity to explore the interplay between proximity-induced
magnetism, spin-orbit coupling and superconductivity. Here we use a combination
of first-principles calculations of NbSe$_{2}$/CrBr$_{3}$ heterostructures and
an analytical theory of Ising superconductivity to analyze the existing
experiments and provide a complete explanation of highly nontrivial and largely
counterintuitive effects: an increase in the magnitude of the superconducting
gap accompanied by the broadening of the tunneling peaks; hysteretic behavior
of the tunneling conductance that sets in $\approx 2$ K below $T_c$; and
nematic symmetry breaking in the superconducting state. The microscopic reason
in all three cases appears to be the interplay between the proximity-induced
exchange splitting and intrinsic defects. Finally, we predict additional
interesting effects that at the moment cannot be addressed experimentally:
spin-filtering when tunneling across CrBr$_{3}$ and tunneling ``hot spots'' in
momentum space that are anticorrelated with regions where the spin-orbit
splitting is maximum.
