Real structures of models of arrangements

We will deal with two “hidden” real structures in the theory of models of subspace arrangements. Given a real subspace arrangement $A$ and its complexification ${\text{A}}_\frac{3}{8}$ , the first structure is a real De Concini-Procesi model $\tilde{Y}_{A}$ that can be seen as the manifold $\operatorname{Re}Y_{{\text{A}}_{\frac{3}{8}}}$ of (canonical) real points inside the complex De Concini-Procesi model $Y_{{\text{A}}_\frac{3}{8}}$ . We will study its combinatorial properties by describing it as a quotient of a real model with corners $CY_\text A$ introduced in 2003. A second structure arises, on the contrary, as an “extension” of $CY_\text A$ , when $A$ is a Coxeter arrangement. We will “add faces” to $CY_\text A$ and obtain a convex body (or even a polytope); this gives rise to an interesting new family of “realized” posets which includes for instance Kapranov's permutoassociahedra.