  The binomial Eulerian polynomials, introduced by Postnikov, Reiner, and
Williams, are $\gamma$-positive polynomials and can be interpreted as
$h$-polynomials of certain flag simplicial polytopes. Recently, Athanasiadis
studied analogs of these polynomials for colored permutations. In this paper,
we generalize them to $\mathbf{s}$-inversion sequences and prove that these new
polynomials have only real roots by the method of interlacing polynomials.
Three applications of this result are presented. The first one is to prove the
real-rootedness of binomial Eulerian polynomials, which confirms a conjecture
of Ma, Ma, and Yeh. The second one is to prove that the symmetric decomposition
of binomial Eulerian polynomials for colored permutations is real-rooted.
Thirdly, our polynomials for certain $\mathbf{s}$-inversion sequences are shown
to admit a similar geometric interpretation related to edgewise subdivisions of
simplexes.
