  We consider the effect of coupling between phonons and a chiral Majorana edge
in a gapped chiral spin liquid with Ising anyons (e.g., Kitaev's non-Abelian
spin liquid on the honeycomb lattice). This is especially important in the
regime in which the longitudinal bulk heat conductivity $\kappa_{xx}$ due to
phonons is much larger than the expected quantized thermal Hall conductance
$\kappa_{xy}^{\rm q}=\frac{\pi T}{12} \frac{k_B^2}{\hbar}$ of the ideal
isolated edge mode, so that the thermal Hall angle, i.e., the angle between the
thermal current and the temperature gradient, is small. By modeling the
interaction between a Majorana edge and bulk phonons, we show that the exchange
of energy between the two subsystems leads to a transverse component of the
bulk current and thereby an {\em effective} Hall conductivity. Remarkably, the
latter is equal to the quantized value when the edge and bulk can thermalize,
which occurs for a Hall bar of length $L \gg \ell$, where $\ell$ is a
thermalization length. We obtain $\ell \sim T^{-5}$ for a model of the
Majorana-phonon coupling. We also find that the quality of the quantization
depends on the means of measuring the temperature and, surprisingly, a more
robust quantization is obtained when the lattice, not the spin, temperature is
measured. We present general hydrodynamic equations for the system, detailed
results for the temperature and current profiles, and an estimate for the
coupling strength and its temperature dependence based on a microscopic model
Hamiltonian. Our results may explain recent experiments observing a quantized
thermal Hall conductivity in the regime of small Hall angle,
$\kappa_{xy}/\kappa_{xx} \sim 10^{-3}$, in $\alpha$-RuCl$_3$.
