If I understand magnetism correctly, then two magnetic fields with opposing directions would repel each other. Could this same affect happen to electromagnetic waves? And if so, could it be reproduced in an experiment?
It is not that magnetic fields repeal each other, but the bodies that are origin of these fields repel if they are oriented in such a way that their fields in the space between them are pointing in opposite directions.
Electromagnetic waves in vacuum obey linear wave equation, so one does not change the other; they pass each other as if the other one did not exist.
In matter, on the other hand, the description of EM field is more complicated, the equation does not need to be linear and there could be phenomenon that would manifest as waves "repelling" or "attracting" each other. Unfortunately, I do not know any specific examples.