  Type IIB supergravity enjoys a discrete non-Abelian duality group, which has
potential quantum anomalies. In this paper we explicitly compute these, and
present the bordism group that controls them, modulo some physically motivated
assumptions. Quite surprisingly, we find that they do not vanish, which naively
would signal an inconsistency of F-theory. Remarkably, a subtle modification of
the standard 10d Chern-Simons term cancels these anomalies, a fact which relies
on the ***specific*** field content of type IIB supergravity. We also discover
other ways to cancel this anomaly, via a topological analog of the
Green-Schwarz mechanism. These alternative type IIB theories have the same low
energy supergravity limit as ordinary type IIB, but a different spectrum of
extended objects. They could either be part of the Swampland, or connect to the
standard theory via domain walls.
