  We describe a construction of geometric U-folds compatible with a non-trivial
extension of the global formulation of four-dimensional extended supergravity
on a differentiable spin manifold. The topology of geometric U-folds depends on
certain flat fiber bundles which encode how supergravity fields are globally
glued together. We show that smooth non-trivial U-folds of this type can exist
only in theories where both the scalar and space-time manifolds have
non-trivial fundamental group and in addition, the scalar map of the solution
is homotopically non-trivial. Consistency with string theory requires smooth
geometric U-folds to be glued using subgroups of the effective discrete
U-duality group, implying that the fundamental group of the scalar manifold of
such solutions must be a subgroup of the latter. We construct simple examples
of geometric U-folds in a generalization of the axion-dilaton model of
$\mathcal{N}=2$ supergravity coupled to a single vector multiplet, whose scalar
manifold is a generally non-compact Riemann surface of genus at least two
endowed with its uniformizing metric. We also discuss the relation between
geometric U-folds and a moduli space of flat connections defined on the scalar
manifold, which involves certain character varieties not studied in the
literature.
