  Stanley introduced a lattice polytope $\mathcal{C}_P$ arising from a finite
poset $P$, which is called the chain polytope of $P$. The geometric structure
of $\mathcal{C}_P$ has good relations with the combinatorial structure of $P$.
In particular, the Ehrhart polynomial of $\mathcal{C}_P$ is given by the order
polynomial of $P$. In the present paper, associated to $P$, we introduce a
lattice polytope $\mathcal{E}_{P}$, which is called the enriched chain polytope
of $P$, and investigate geometric and combinatorial properties of this
polytope. By virtue of the algebraic technique on Gr\"{o}bner bases, we see
that $\mathcal{E}_P$ is a reflexive polytope with a flag regular unimodular
triangulation. Moreover, the $h^*$-polynomial of $\mathcal{E}_P$ is equal to
the $h$-polynomial of a flag triangulation of a sphere. On the other hand, by
showing that the Ehrhart polynomial of $\mathcal{E}_P$ coincides with the left
enriched order polynomial of $P$, it follows from works of Stembridge and
Petersen that the $h^*$-polynomial of $\mathcal{E}_P$ is $\gamma$-positive.
Stronger, we prove that the $\gamma$-polynomial of $\mathcal{E}_P$ is equal to
the $f$-polynomial of a flag simplicial complex.
