Mutual statistics between dyons (charge-monopole composite)

I am asking for some intuitive understanding between two dyons with $$(e,m)$$ in 3-dimensional space. Here the magnetic charge $$m$$ is normalized as $$\begin{eqnarray} m=\int_{S^2}\frac{B}{2\pi}\in\mathbb{Z}, \end{eqnarray}$$ where $$S^2$$ is a small sphere around the dyon and $$B$$ is its magnetic field strength.

It was shown that the mutual statistics or the phase related to exchanging such two dyons is $$\begin{eqnarray} \exp(i\theta)=(-1)^{em}. \end{eqnarray}$$ My understanding is that for two dyons with their own magnetic flux restricted to an infinitesimal solid angle, we can calculate the half of Aharonov-Bohm phase when one dyon encircles the other as the statistics angle above. However, this argument strongly depends on the flux distribution because the phase will be halved if we let the monopole magnetic flux distribute spherically symmetrically around the dyon.

My question is how to understand this dyon statistics in an intuitive way and what is the mistake of my argument above?