  We compute the contribution of phonons to the shear viscosity $\eta$ in
superfluid neutron stars, assuming neutron pairing in a $^1S_0$ channel. We use
a Boltzmann equation amended by a collision term that takes into account the
binary collisions of phonons. We use effective field theory techniques to
extract the phonon scattering rates, written as a function of the equation of
state (EoS) of the system. Our formulation is rather general, and can be used
to extract the shear viscosity due to binary collisions of phonons for other
superfluids, such as the cold Fermi gas in the unitarity limit. We find that
$\eta \propto 1/T^5$, the proportionality factor depending on the EoS of the
system. Our results indicate that the phonon contribution to $\eta$ cannot be
ignored and might have relevant effects in the dynamics of the different
oscillation modes of the star.
