Phase term in Aharonov-Bohm effect

In $U(1)$ gauge, the transformation is given by $(c=1)$ $$e^{frac{ieint A_mu dx_mu}{hbar}}$$ I know that this form comes from the phase picked up by electrons in Aharonov-Bohm effect. However, in the magnetic Aharonov-Bohm effect, the form is connected with the existence of magnetic monopole. Such a monopole would restore a symmetry between the electric and magnetic sources in Maxwell’s equations.

My question is if a monopole is never found, does that imply a modification in the phase term? What will be the consequence if the phase changes by a constant factor independent of $$x_mu?$$

Your comments are highly appreciated.