  A classic problem connecting algebraic and geometric combinatorics is the
realization problem: given a poset, determine whether there exists a polytope
whose face lattice is the poset. In 1990s, Kapranov defined a poset as a hybrid
between the face poset of a permutohedron and that of an associahedron, and he
asked whether this poset is realizable. Shortly after his question was posed,
Reiner and Ziegler provided a realization. Based on our previous work on the
nested braid fan, we provide in this paper a different realization of
Kapranov's poset by constructing the vertex set and the normal fan of a
permuto-associahedron simultaneously.
