The positive Bergman complex of an oriented matroid

We study the positive Bergman complex B(M) of an oriented matroid M , which is a certain subcomplex of the Bergman complex B(M) of the underlying unoriented matroid M . The positive Bergman complex is defined so that given a linear ideal I with associated oriented matroid MI , the positive tropical variety associated to I is equal to the fan over B(MI). Our main result is that a certain “fine” subdivision of B (M) is a geometric realization of the order complex of the proper part of the Las Vergnas face lattice of M . It follows that B(M) is homeomorphic to a sphere. For the oriented matroid of the complete graph Kn, we show that the face poset of the “coarse” subdivision of B (Kn) is dual to the face poset of the associahedron An−2, and we give a formula for the number of fine cells within a coarse cell.