One approach to the Strong CP Problem (known as Nelson-Barr models) is to assume that parity is a gauge symmetry, which is spontaneously broken in the world around us. In this talk, we argue that the associated domain walls are exactly stable as the result of a topological gauge charge. We explain how these charges are compatible with the expected absence of global symmetries in quantum gravity as well as their relationship with the Swampland Cobordism Conjecture.