  F-theory models are constructed where the 7-brane has a non-trivial
fundamental group. The base manifolds used are a toric Fano variety and a
smooth toric threefold coming from a reflexive polyhedron. The discriminant
locus of the elliptically fibered Calabi-Yau fourfold can be chosen such that
one irreducible component it is not simply connected (namely, an Enriques
surface) and supports a non-Abelian gauge theory.
