  We discuss gauge theories of the Yang-Mills kind in finite regions with
boundaries, and in particular the definition of the corresponding quasi-local
degrees of freedom and their gluing upon composition of the underlying regions.
Although the most of the technical results presented here has appeared in
previous works by Gomes, Hopfm\"uller and the author, we adopt here a new
perspective. Focusing on Maxwell theory as our model theory, in most of the
text we avoid technical complications and focus on the conceptual issues
related to symplectic reduction in finite and bounded regions, and to
gluing$-$e.g. superselection sectors, non-locality, Dirac's dressing of charged
fields, and edge modes. In this regard, the title refers to a gluing formula
for the reduced symplectic structures, where the "edge mode" contribution is
explicitly computed in terms of gauge-invariant bulk variables. Despite
capturing most interesting features, the Abelian theory misses some crucial
technical and conceptual points which are present in the non-Abelian case. To
fill this gap, we dedicate the last section to a brief overview of functional
connection forms, flux rotations, and geometric BRST, among other topics.
