Although several extensions to the Standard Model predict the possible existence of magnetic monopoles, their expected properties are rather significantly different from those of the electrically-charged particles we all know and love--the coupling constant for monopoles is huge, if they exist they are experimentally constrained to huge masses, and as such would not form interesting complex structures like normal baryonic matter held together by electric charges.
It would seem to me more "elegant" if electromagnetism were in fact fully symmetric, such that charged leptons & quarks all come in electric / magnetic pairs, and coupling constants and masses were more similar such that, although the results need not be absolutely identical, both versions of matter can form something like atoms and molecules, etc.
So, in the spirit of thought experiments like A Universe Without Weak Interactions, or Greg Egan's alternate-metric universes (Dichronauts & Orthogonal), is it possible to construct a fully symmetric theory with co-existent electric and magnetic matter? How off-the-rails do things get if we just set $\alpha$ to 1 (so it's reciprocal is also 1)? If it is possible, what are the minimal other changes necessary to preserve a universe with recognizable chemistry? And given the existence of parallel electric and magnetic quarks, would electric and magnetic baryons still exhibit strong force interactions with each other, resulting in dyonic nucleii, or do we end up with two completely isolated parallel periodic tables?