  There are several real spherical models associated with a root arrangement,
depending on the choice of a building set. The connected components of these
models are manifolds with corners which can be glued together to obtain the
corresponding real De Concini-Procesi models. In this paper, starting from any
root system Phi with finite Coxeter group W and any W-invariant building set,
we describe an explicit realization of the real spherical model as a union of
polytopes (nestohedra) which lie inside the chambers of the arrangement. The
main point of this realization is that the convex hull of these nestohedra is a
larger polytope, a permutonestohedron, equipped with an action of W or also,
depending on the building set, of Aut(Phi). The permutonestohedra are natural
generalizations of Kapranov's permutoassociahedra.
