Magnetic monopole, vector-potential and differential forms

When written in the language of exterior algebra, Maxwell-Thomson equation writes as $dB=0$ where $d$ is the exterior derivative and $B$ is the magnetic flux 2-form. From Poincaré’s lemma, it follows that as $B$ is closed, it is locally exact. Depending on the topology of the domain of interest, $B$ derives from a vector potential 1-form according to $B=dA$.

In Manton and Sutcliffe’s Topological Solitons chapter 8, the existence of magnetic monopoles is discussed in the framework of vector analysis and piecewise-defined vector potentials $vec A_{pm}$ are derived. As they differ from a gauge term (namely $vec A_-=vec A_+-vecnablaleft(frac{g}{2pi}phiright)$), they describe the same physical field.

But when translated into the language of forms, there is something that looks paradoxical. Indeed, as the exterior derivative is a nilpotent operator $d^2equiv 0$, I wonder how one can simultaneously have $dB=d^2 A=0$ on the one hand and, due to the presence of magnetic monopoles, $dBneq 0$ on the other hand. I suspect the solution lies in some topological subtleties, but I cannot see what they are. Can anyone help me understand the problem please?