  Viscous unified dark matter models aim to describe the dark sector of the
Universe, as the dark matter fluid itself gives rise to an accelerated
expansion due to a negative bulk viscous pressure. However, in most studies,
radiation is often disregarded as a minor factor in dynamical system analyses,
overlooking whether radiation domination is achievable. In this paper, we
rigorously examine this critical aspect for common parameterizations of bulk
viscosity, denoted as $\xi$, within two general classes of viscous unified
models. Our findings reveal significant inconsistencies in models where $\xi
\propto H^{1-2s} \rho_{m}^{s}$ with $s\leq 0$, and surprisingly, in models
where $\xi \propto \rho_{m}^{s}$ with exponents $s<0$, as they both fail to
produce a radiation-dominated era. Moreover, the exponent $s$ must lye within
the interval $0 \leq s < 1/2$ for the latter model to correctly describes the
cosmological evolution. These results underscore the need of including these
constraints as a prior in statistical analyses of observational data, with
implications for current statistical inferences for the second model, where
both prior and best-fit values of $s$ often fall outside the acceptable range.
