In this question, I'll define a gauge symmetry to be a symmetry that is merely a redundancy of our description of the system. Typically, gauge symmetries are also local symmetries, and a global symmetry can be made into a local gauge symmetry by introducing a covariant derivative and a gauge field.

But what about the case of turning a global symmetry into a *global* gauge symmetry? For example, the Schrodinger equation has a global $U(1)$ symmetry by multiplying the wavefunction by phases. This global symmetry is typically gauged, i.e. we regard vectors in the Hilbert space as physically equivalent if they differ by a phase.

However, I've never heard of other cases of this. Are there situations where such a global gauge symmetry would be useful, or is there some reason this isn't useful?