In the chiral limit, the long distance effective theory of QCD at finite temperature is not hydrodynamics but a kind of non-abelian superfluid hydrodynamics. We describe this theory and its viscous corrections, including also a correction due to the finite quark mass. At finite quark mass, the long distance theory is ordinary hydrodynamics, and the superfluid theory then just determines non-analytic in the quark mass corrections to the transport coefficients of QCD, akin to the "long time tails" of hydro. We show how this works out for the bulk viscosity. In chiral perturbation theory the dissipative parameters of the superfluid theory can be computed diagrammatically, and we do this. These results then determine the leading order the bulk viscosity of the pion gas close to the chiral limit.