It is well known that Maxwell equations can be made symmetric w.r.t. $E$ and $B$ by introducing non-zero magnetic charge density/flux.
In this case we have $div B = \rho_m$, where $\rho_m$ is a magnetic charge density.
But this means that $B$ cannot be expressed as the curl of vector potential anymore.
Does it mean that it is impossible to develop electrodynamics of non-zero electric and magnetic charges in terms of vector and scalar potentials? What happens to $U(1)$ gauge invariance then?
P.S.: I know that for the point-like magnetic charge vector potential can still be introduced, but not in the entire space. My question is related to non point-like magnetic charge densities.