On quadrangulations and Stokes complexes

Stokes complexes consist of sets of mutually noncrossing diagonals of a convex polygon, that are in some sense compatible with a reference quadrangulation. Originally defined by Y. Baryshnikov (2001), they were recently revisited by F. Chapoton (2016) who proposed several conjectures. We settle two of these conjectures and study geometric realizations of Stokes complexes using compatibility vectors.