  We apply a recently developed first-principles based approach for treating
generic spin-phonon couplings in materials with strong spin-orbit coupling to
study $\alpha$-RuCl$_3$. Of particular focus is the potential for this material
to exhibit a phonon thermal Hall effect induced by spin-phonon interactions. We
find that spin-orbit coupling significantly enriches the form of these
interactions, and imbues them with chirality that is conducive to generating
finite phonon Berry curvatures. We show that this leads to a phonon thermal
Hall effect that qualitatively reproduces the measured field dependence of
$\kappa_{xy}$ without requiring a field-induced spin liquid.
