Hey Brian, I guess it depends on what comes first. The way I see it, Schrödinger's equation has a U(1) degree of freedom (ie the electric charge). Introducing the covariant derivative (ie relativity), a rank-2 antisymmetric tensor is necessary to ensure gauge invariance. One representation of this tensor is the electromagnetic tensor with a symmetric (E) and an anti-symmetric (B) components, from which one derives Maxwell's equations.

See also this:

https://nicf.net/articles/classical-em/