 Solution

A gauge redundancy relates different field representatives of the same physical configuration. For example, the transformation just exhibited changes $A_{\mu }$ and the scalar phase but leaves $F_{\mu \nu }$, $|\phi {|}^2$ and all other gauge-invariant operators unchanged. Fixing a gauge removes this freedom; it does not remove a physical symmetry-related state. In canonical quantization, local transformations trivial at the boundary are generated by the Gauss law constraint in gauge theory, and physical states satisfy that constraint.

A global symmetry in field theory instead has a spacetime-independent parameter and can act nontrivially on physical states. For a continuous global symmetry in field theory, Noether theorem gives a conserved charge that generates the transformation and labels the states; discrete global symmetries do not acquire a Noether current from the same infinitesimal theorem. These are statements about physical transformations, rather than merely whether a parameter is written as a constant or a function.

Boundary conditions matter in gauge theory. gauge transformations that vanish at spatial infinity are redundancies, while transformations nontrivial at infinity may act through a physical electric charge. In particular a constant phase transformation can survive as a global charge symmetry rather than being quotiented out with the local gauge redundancy. Thus local gauge redundancy is a redundancy of description; genuine global symmetry in field theory can distinguish physical states and charges, with the boundary qualification made explicit.