# Effect of introducing magnetic charge on use of vector potential [duplicate]

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.

Although in the presence of magnetic monopoles $\mathbf{B}$ can no longer be expressed as the curl of a vector potential, it can still be written as the sum of the curl of a vector potential and the gradient of a scalar potential:
$\mathbf{B} = \nabla\Xi + \nabla\times \mathbf{A}$
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