  We study the phases of the SU(N1)X SU(N2) gauge theory with a bi-fundamental
fermion in 3+1 dimensions. We show that the discrete anomalies and Berry phases
associated to the one-form symmetry of the theory allow for several
topologically distinct phase diagrams. We identify several limits of the theory
where the phase diagram can be determined using various controlled
approximations. When the two ranks are equal N1=N2, these limits all lead to
the same topology for the phase diagram and provide a consistent global
understanding of the phases of the theory. When the ranks are different,
different limits lead to distinct topologies of the phase diagram. This
necessarily implies non-trivial physics at some intermediate regimes of
parameter space. In the large N1,N2 limit, we argue that the topological
transitions are accounted for by a (non-supersymmetric) duality cascade as one
varies the parameters of the theory.
