I would like to know what are the formal rules for two-player bughouse without a clock. I'd like to think of each team as a single player.
It seems to me that bughouse chess works primarily because of the time limit imposed by the clock. If one should remove the clock, then it is easy to imagine a position where both teams are at a disadvantage to move first. Perhaps it is one team's turn to move on board 1 and the other team's turn on board 2, but whoever moves first will be put at a disadvantage on the other board because of a resulting capture. In such a case, neither player wants to go first, and the game enters a deadlock, with no team specifically to blame for not having moved.
Are there formal rule sets for bughouse chess that eliminate the possibility of such a deadlock?
I suppose that if one insists that play proceeds on each board in turn (white on board 1, black on board 2, white on board 2, black on board 1, repeat), then it would resolve this issue, since it would at any stage be a certain player's obligation to move, just as in any game. But in practice, bughouse chess does not obey such a convention, and I wonder whether there is a way to set up the rules to more closely follow the practice, but while avoiding the problematic kind of deadlock where neither team wants to move, but nobody is obliged to move.