  We give a global formulation of the coupling of four-dimensional scalar sigma
models to Abelian gauge fields for the generalized situation when the "duality
structure" of the Abelian gauge theory is described by a flat symplectic vector
bundle $(\mathcal{S},D,\omega)$ defined over the scalar manifold $\mathcal{M}$.
The construction uses a taming of $(\mathcal{S}, \omega)$, which encodes
globally the inverse gauge couplings and theta angles of the "twisted" Abelian
gauge theory in a manner that makes no use of duality frames. We show that
global solutions of the equations of motion of such models give classical
locally geometric U-folds. We also describe the groups of duality
transformations and scalar-electromagnetic symmetries arising in such models,
which involve lifting isometries of $\mathcal{M}$ to a particular class of flat
automorphisms of the bundle $\mathcal{S}$ and hence differ from expectations
based on local analysis. The appropriate version of the Dirac quantization
condition involves a discrete local system defined over $\mathcal{M}$ and gives
rise to a smooth bundle of polarized Abelian varieties, endowed with a flat
symplectic connection. This shows that a generalization of part of the
mathematical structure familiar from $\mathcal{N}=2$ supergravity is already
present in such purely bosonic models, without any coupling to fermions and
hence without any supersymmetry.
