  The recent observation of a half-integer quantized thermal Hall effect in
$\alpha$-RuCl$_3$ is interpreted as a unique signature of a chiral spin liquid
with a Majorana edge mode. A similar quantized thermal Hall effect is expected
in chiral topological superconductors. The unavoidable presence of gapless
acoustic phonons, however, implies that, in contrast to the quantized
electrical conductivity, the thermal Hall conductivity $\kappa_xy$ is never
exactly quantized in real materials. Here, we investigate how phonons affect
the quantization of the thermal conductivity focusing on the edge theory. As an
example we consider a Kitaev spin liquid gapped by an external magnetic field
coupled to acoustic phonons. The coupling to phonons destroys the ballistic
thermal transport of the edge mode completely, as energy can leak into the
bulk, thus drastically modifying the edge-picture of the thermal Hall effect.
Nevertheless, the thermal Hall conductivity remains approximately quantized and
we argue, that the coupling to phonons to the edge mode is a necessary
condition for the observation of the quantized thermal Hall effect. The
strength of this edge coupling does, however, not affect the conductivity. We
argue that for sufficiently clean systems the leading correction to the
quantized thermal Hall effect, $\Delta \kappa_{xy}/T \sim \text{sign(B)} \,
T^2$, arises from a intrinsic anomalous Hall effect of the acoustic phonons due
to Berry phases imprinted by the chiral (spin) liquid in the bulk. This
correction depends on the sign but not the amplitude of the external magnetic
field.
