  A permutation $\pi$ of a multiset is said to be a {\em quasi-Stirling}
permutation if there does not exist four indices $i<j<k<\ell$ such that
$\pi_i=\pi_k$ and $\pi_j=\pi_{\ell}$. For a multiset $\mathcal{M}$, denote by
$\overline{\mathcal{Q}}_{\mathcal{M}}$ the set of quasi-Stirling permutations
of $\mathcal{M}$. The {\em qusi-Stirling polynomial} on the multiset
$\mathcal{M}$ is defined by $ \overline{Q}_{\mathcal{M}}(t)=\sum_{\pi\in
\overline{\mathcal{Q}}_{\mathcal{M}}}t^{des(\pi)}$, where $des(\pi)$ denotes
the number of descents of $\pi$. By employing generating function arguments,
Elizalde derived an elegant identity involving quasi-Stirling polynomials on
the multiset $\{1^2, 2^2, \ldots, n^2\}$, in analogy to the identity on
Stirling polynomials. In this paper, we derive an identity involving
quasi-Stirling polynomials $\overline{Q}_{\mathcal{M}}(t)$ for any multiset
$\mathcal{M}$, which is a generalization of the identity on Eulerian polynomial
and Elizalde's identity on quasi-Stirling polynomials on the multiset $\{1^2,
2^2, \ldots, n^2\}$. We provide a combinatorial proof the identity in terms of
certain ordered labeled trees. Specializing $\mathcal{M}=\{1^2, 2^2, \ldots,
n^2\}$ implies a combinatorial proof of Elizalde's identity in answer to the
problem posed by Elizalde. As an application, our identity enables us to show
that the quasi-Stirling polynomial $\overline{Q}_{\mathcal{M}}(t)$ has only
real roots and the coefficients of $\overline{Q}_{\mathcal{M}}(t)$ are unimodal
and log-concave for any multiset $\mathcal{M}$, in analogy to Brenti's result
for Stirling polynomials on multisets.
