I don’t think so. The prisoner’s dilemma is just a payoff matrix. Adding communication doesn’t change it as long as each player can’t see the other’s choice before making their own. And whether the payoff matrix arises because of a mutual adversary or some other way doesn’t seem like it should make a difference to me.
Make it an iterated prisoner’s dilemma, or add some other way for players to punish each other after seeing their choice, and that does change the game.
Make it an iterated prisoner’s dilemma, or add some other way for players to punish each other after seeing their choice, and that does change the game.