  Gauge theories possess nonlocal features that, in the presence of boundaries,
inevitably lead to subtleties. We employ geometric methods rooted in the
functional geometry of the phase space of Yang-Mills theories to: (1)
characterize a basis for quasilocal degrees of freedom (dof) that is manifestly
gauge-covariant also at the boundary; (2) tame the non-additivity of the
regional symplectic forms upon the gluing of regions; and to (3) discuss gauge
and global charges in both Abelian and non-Abelian theories from a geometric
perspective. Naturally, our analysis leads to splitting the Yang-Mills dof into
Coulombic and radiative. Coulombic dof enter the Gauss constraint and are
dependent on extra boundary data (the electric flux); radiative dof are
unconstrained and independent. The inevitable non-locality of this split is
identified as the source of the symplectic non-additivity, i.e. of the
appearance of new dof upon the gluing of regions. Remarkably, these new dof are
fully determined by the regional radiative dof only. Finally, a direct link is
drawn between this split and Dirac's dressed electron.
