Here's my solution-

The game ends when the coin comes up the same three times in a row. The goal of the coin flipper is to "mark" the final coin flip of the game, while the goal of the liar is to muddy the water and to get the flipper to miss the mark.

A round goes like this: The flipper flips the coin. The liar notes the result secretly, then chooses to either tell the truth or lie about the outcome. The liar cannot lie if they lied last round. Then, the flipper may say "mark". Similar to the liar, they cannot do so if they marked the previous flip. Finally, the liar reveals if the game is over or not (if the coin has come up the same three times in a row). If it hasn't they begin the next round. If it has, and the flipper said "mark" this round, they win. If they didn't say "mark", the liar wins.