I don't think it's a big deal to see a kid acting out when in public. It happens to everyone that deals with children at some point or another (or multiple times lol). I would go about it the same way you would if it was happening at home/dc. Children aren't dumb and figure out real quick how and when to play the game if we aren't consistent with them. Don't worry about what others think and just congratulate yourself in being strong enough to enforce the rules no matter what. I know it's terrible to judge just a glimpse of someone's elses life but it drives me crazy when I see an adult giving into a child just so they don't tantrum in public.
Knock on wood.....my dck's have never tantrumed in public though. I usually get told by multiple people how well behaved my children are (they always assume I'm the mom not the dcprovider). I have 2 in my care now that will probably change this though!! haha.