If you have attacked and damaged a mystics character on your turn, then ended your turn, but your opponent did not call for mystics damage, can they ask you to take the mystics damage halfway through their next turn?

I would say no. Unless you want to be nice.

When a character using the Mystics team ability takes damage from an attack, the attacker is dealt 1 unavoidable damage. This damage is not an attack.

There is no "may" in that ability; it is not optional. He needed to take that damage.

Misplays do occur, though, so once it was missed everyone should just carry on and try to remember the next time.

However, and this is a big "however", BOTH players are required to help maintain the state of play. If your opponent attacked your character and knew it was a Mystics, he should have clicked his character anyway. He was just as responsible for playing the required effect correctly. If he knew about the TA and didn't do anything just because he was hoping you'd forget, he was cheating.

Mystics is not optional!

Both players are responsible for remembering that the damage has to be taken.

Sometimes both players forget, it was unintentional but well after the fact. While I agree that both players "should" keep track and try to remember these things, let's be honest, with the extreme amount of thing we are now forced to remember on a game to game basis these things can happen and if it happens in a big tournament then what? What is an acceptable amount of rounds you should be allowed to go back when you remember something that was forgotten by both players? "Oh, you had a minus 1 to your attack last round from Nightmare, that attack would have missed so I'll just heal my guy that damage I took from your attack because we both forgot." Really?

That is why there are Volunteer/Judge/Venue running the event. They should be called over and asked. They would make that decision.

Ok thanks, Judge call makes sense to me.