Classically there is no problem with defining Maxwell's for magnetic monopoles. They look like (in tensor notation): $$ \partial_\mu F^{\mu\nu} = J_e^{\nu} $$ $$ \partial_\mu* F^{\mu\nu} = J^\nu_m$$ Here $F^{\mu\nu}$ is the Faraday tensor, and $J^\nu_e$, $J^\nu_m$ are respectively the electric and magnetic four currents. Also, $*$ is the hodge star operator. These are well defined as PDE's, which means you can use standard PDE methods to analyze them. For example, we can consider the magnetostatic case of a single point charge as in Habouz answer.

However, there are also additional complications that arise in theories with magnetic monopoles. One is that you cannot write things In terms of potentials in a simple way. This is because $\partial_\mu *(\partial^{[\mu} A^{\nu]})= 0$ which means that writing $F^{\mu\nu} = \partial^{[\mu} A^{\nu]}$ would be inconsistent with our second maxwell equation in the presence of magnetic monopoles. There are ways to get around this such as the Dirac string, but they are all rather more complicated than regular electrodynamics. This matters because describing things in terms of potentials is how we get local couplings in quantum theories (consider the Aharanov Bohm effect).

In general, the quantum theory for magnetic monopoles is much more complicated. For example, we find that to have a consistent theory we must have a charge quantization condition. Many consider this a positive aspect of the theory because it gives a possible explanation for why we observe charge to be only found in integer multiples of some fundamental charge.

In particular, we find that the magnetic coupling constant is inversely proportional to the electric. So, as you mentioned, the electric coupling is very small, and the magnetic is big. This makes it very hard to use traditional quantum field theory methods because they are generally based on perturbation theory.

To summarize, as far as I know, there are no issues with the classical theory of magnetic monopoles i.e. we can analyze Maxwell's equations with magnetic currents using standard PDE methods. They are more complicated because we have added extra inhomogeneity, but they are perfectly well-defined (and lead to an inverse square law for example). It is moving to the corresponding quantum field theory where things become much more complicated.

Classically there is no problem with defining Maxwell's equations for magnetic monopoles. They look like (in tensor notation): $$ \partial_\mu F^{\mu\nu} = J_e^{\nu} $$ $$ \partial_\mu* F^{\mu\nu} = J^\nu_m$$ Here $F^{\mu\nu}$ is the Faraday tensor, and $J^\nu_e$, $J^\nu_m$ are respectively the electric and magnetic four currents. Also, $*$ is the hodge star operator. These are well defined as PDE's, which means you can use standard PDE methods to analyze them. For example, we can consider the magnetostatic case of a single point charge as in Habouz answer.

However, there are also additional complications that arise in theories with magnetic monopoles. One is that you cannot write things In terms of potentials in a simple way. This is because $\partial_\mu *(\partial^{[\mu} A^{\nu]})= 0$ which means that writing $F^{\mu\nu} = \partial^{[\mu} A^{\nu]}$ would be inconsistent with our second maxwell equation in the presence of magnetic monopoles. There are ways to get around this such as the Dirac string, but they are all rather more complicated than regular electrodynamics. This matters because describing things in terms of potentials is how we get local couplings in quantum theories (consider the Aharanov Bohm effect).

In general, the quantum theory for magnetic monopoles is much more complicated. For example, we find that to have a consistent theory we must have a charge quantization condition. Many consider this a positive aspect of the theory because it gives a possible explanation for why we observe charge to be only found in integer multiples of some fundamental charge.

In particular, we find that the magnetic coupling constant is inversely proportional to the electric. So, as you mentioned, the electric coupling is very small, and the magnetic is big. This makes it very hard to use traditional quantum field theory methods because they are generally based on perturbation theory.

To summarize, as far as I know, there are no issues with the classical theory of magnetic monopoles i.e. we can analyze Maxwell's equations with magnetic currents using standard PDE methods. They are more complicated because we have added extra inhomogeneity, but they are perfectly well-defined (and lead to an inverse square law for example). It is moving to the corresponding quantum field theory where things become much more complicated.