  It has become common knowledge that phonons can generate thermal Hall effect
in a wide variety of materials, although the underlying mechanism is still
controversial. We study longitudinal $\kappa_{xx}$ and transverse $\kappa_{xy}$
thermal conductivity in Pr$_2$Ir$_2$O$_7$, which is a metallic analogue of spin
ice. Despite the presence of mobile charge carriers, we find that both
$\kappa_{xx}$ and $\kappa_{xy}$ are dominated by phonons. A $T/H$ scaling of
$\kappa_{xx}$ unambiguously reveals that longitudinal heat current is
substantially impeded by resonant scattering of phonons on paramagnetic spins.
Upon cooling, the resonant scattering is strongly affected by a development of
spin ice correlation and $\kappa_{xx}$ deviates from the scaling in an
anisotropic way with respect to field directions. Strikingly, a set of the
$\kappa_{xx}$ and $\kappa_{xy}$ data clearly shows that $\kappa_{xy}$
correlates with $\kappa_{xx}$ in its response to magnetic field including a
success of the $T/H$ scaling and its failure at low temperature. This
remarkable correlation provides solid evidence that an indispensable role is
played by spin-phonon scattering not only for hindering the longitudinal heat
conduction, but also for generating the transverse response.
