  Comparison of hydrodynamic and "hybrid" hydrodynamics+transport calculations
to heavy-ion data inevitably requires the conversion of the fluid to particles.
For dissipative fluids the conversion is ambiguous without additional theory
input complementing hydrodynamics. We obtain self-consistent shear viscous
phase space corrections from linearized Boltzmann transport theory for a gas of
hadrons. These corrections depend on the particle species, and incorporating
them in Cooper-Frye freezeout affects identified particle observables. For
example, with additive quark model cross sections,proton elliptic flow is
larger than pion elliptic flow at moderately high $p_T$ in $Au+Au$ collisions
at RHIC. This is in contrast to Cooper-Frye freezeout with the commonly used
"democratic Grad" ansatz that assumes no species dependence. Various analytic
and numerical results are also presented for massless and massive two-component
mixtures to aid the interpretation. Self-consistent viscous corrections for
each species are tabulated in Appendix F for convenient inclusion in pure
hydrodynamic and hybrid calculations.
