Add to your theory/model. In order to make the symmetry work, you need something to "fix" the discrepancy you would otherwise have in the equations. So you add

Add to your theory/model. In order to make the symmetry work, you need something to "fix" the discrepancy you would otherwise have in the equations. So you add this field (variable) to make the whole thing consistent. In the end, you see you needed this "gauge field" (for historical reasons these are called "gauge symmetries") which is electromagnetism.