  Colored multiset Eulerian polynomials are a common generalization of
MacMahon's multiset Eulerian polynomials and the colored Eulerian polynomials,
both of which are known to satisfy well-studied distributional properties
including real-rootedness, log-concavity and unimodality. The symmetric colored
multiset Eulerian polynomials are characterized and used to prove sufficient
conditions for a colored multiset Eulerian polynomial to be self-interlacing.
The latter property implies the aforementioned distributional properties as
well as others, including the alternatingly increasing property and
bi-$\gamma$-positivity. To derive these results, multivariate generalizations
of an identity due to MacMahon are deduced. The results are applied to a pair
of questions, both previously studied in several special cases, that are seen
to admit more general answers when framed in the context of colored multiset
Eulerian polynomials. The first question pertains to $s$-Eulerian polynomials,
and the second to interpretations of $\gamma$-coefficients.
