"Should martial characters be mundane, realistic, or supernatural?"
Because I think many are thinking mundane = realistic as posters have suggested John Wick as mundane.
I think the word you are looking for is extraordinary. I think it fine to make subclasses to do that, and some like Cavalier, Champion and Battlemaster already do things like that a bit, albeit in small doses. I would be totally cool with new subclasses of Fighter, Barbarian, Rogue and Monk to explore those concepts. I would also be cool with feats that accomplish these sorts of things.
That said, I think the classes themselves already have as much as they need in that regard through the rules in the PHB and DMG and I would be opposed to any changes in the classes themselves to accomplish this, especially since if you put that in the class mechanics you get it AND the things, often supernatural, that you get through the subclasses.
If you said the exact "realistic" things you want to do it would be easier to understand