  This paper investigates isomorphisms of Bergman fans of matroids respecting
different fan structures, which we regard as matroid analogs of birational
maps. We show that isomorphisms respecting the fine fan structure are induced
by matroid isomorphisms. We introduce Cremona automorphisms of the coarse
structure of Bergman fans, which are not induced by matroid automorphisms. We
show that the automorphism group of the coarse fan structure is generated by
matroid automorphisms and Cremona maps in the case of rank 3 matroids which are
not parallel connections and for modularly complemented matroids.
