I'm quite certain that if I painted my house white and red with silver curly letters on it spelling "Cola is Sugar" there'd be multiple laws that expensive lawyers can help enforce.

Those laws (trademark, libel, etc) practically grant large corporations from taking over public space. In my country there's committees (seated by advertising industry) that monopolize the space and enforce this in even greater detail. Hell, we even have committees that tell what color you are allowed to paint your front-door (Schoonheidscommitee). This latter, however, is democratically ruled (local govts) so something that Coca Cola has no seat in.

So the answer to your question is: yes. They already can, while the opposite is not possible.

But the question itself is actually a false dichotomy. Me being able to oppose Coca Cola on their own turf: by running a campaign against them, does not imply that Coca Cola can automatically then decide the color of my house.