  We introduce a positive definite single-particle distribution that is
suitable for describing the transition from a macroscopic hydrodynamic to a
microscopic kinetic description during the late stages of heavy-ion collisions
in the presence of moderately large viscous corrections. The modified
equilibrium distribution function can be constructed with hydrodynamic input
from either relativistic viscous fluid dynamics or anisotropic fluid dynamics.
We test the modified equilibrium distribution's hydrodynamic output for a
stationary hadron resonance gas subject to either shear stress, bulk pressure,
or baryon diffusion current at a given freeze-out temperature and baryon
chemical potential. While it does not reproduce all components of the net
baryon current and energy-momentum tensor exactly, it significantly improves
upon the customary linearized approximations for the non-equilibrium correction
$\delta f_n$ which typically lead to unphysical negative distribution functions
at large particle momenta. A comparison of particle spectra and
$p_T$-differential elliptic flow coefficients from the Cooper-Frye formula
computed with the modified equilibrium distribution and with linearized $\delta
f_n$ corrections is presented, for two different (2+1)-dimensional
hypersurfaces corresponding to central and non-central Pb+Pb collisions at the
Large Hadron Collider (LHC).
