Magnetic monopoles and special relativity

The mathematical model for classical electromagnetism just doesn't forbid magnetic monopoles by construction.

Consider an arbitrary vector field $X$ in 3d. Such a vector field is totally characterized by its divergence and curl. Suppose the following is true:

$$\nabla \cdot X = \sigma, \quad \nabla \times X = Y$$

Then knowing $\sigma$ and $Y$ everywhere, one can reconstruct $X$ everywhere. The scalar field $\sigma$ and the vector field $Y$ are "sources" of $X$. That's important: it means that one only has so much freedom to add sources for a vector field.

Magnetic monopoles are one such freedom. In the context of EM, we know $\nabla \cdot B = 0$ from observations, but we still model $B$ as a vector field, and people still wonder what would happen if $B$ had nonzero divergence instead (corresponding to a magnetic monopole). This is something the math allows even though it might not be physical reality.

Such issues are very common in physics, though it's something of a matter of point of view. Why should we consider $B$ not being divergenceless, instead of, say, $B$ not being exactly a vector field? There's no real answer to that--it's just that physicists routinely must poke at the boundaries of a model to see if there's a prediction that might not hold, to see if there's a measurement or experimental result that could be confirmed more precisely.

Magnetic monopoles are just relatively easy to add to the theory.

The idea that magnetism is a side-effect of electricity is deeply mistaken. The sooner you forget about that idea, the better.

Electricity and magnetism are the two aspects of a single phenomenon, electromagnetism. Read that sentence again: They are not two aspects of electricity, they are two aspects of electromagnetism. Electricity does not cause magnetism and magnetism does not cause electricity, but rather special relativity joins them together, just as it joins space and time into spacetime. (See also my related answer here).

For most electromagnetic phenomena, there is no frame of reference in which it is purely electric, or purely magnetic. It is always a mix of the two, although it's a different mix in different frames of reference. For example, there is no frame of reference in which a refrigerator magnet has no magnetic field. In some frames it will also have an electric field, but a magnetic field is there too.

So in conclusion, magnetism is a fundamental physical phenomenon in its own right, not merely a funny way of talking about certain aspects of electricity. For that reason, it's possible for magnetism to contain phenomena that cannot be directly extrapolated from everyday electricity plus SR.

PS: If there are no magnetic monopoles in a certain reference frame, then there are no magnetic monopoles in ANY reference frame. Conversely, if there ARE magnetic monopoles in one reference frame, then there are magnetic monopoles in every reference frame.