When both players resign

"The game is won by the player whose opponent declares he resigns. This immediately ends the game." This is section 5.1.2 in the FIDE Laws of Chess. At some point in the past it was suggested that two players could scam the system by both resigning simultaneously, thereby earning each player a full point. The FIDE Rules Commission even discussed this (briefly), and IA Franca Dapiran made the sensible suggestion to only accept the resignation of the player who had the move.

Of course such a bizarre situation would not happen in practice now, would it? Well, not exactly.

At Street Chess today something awfully close to this did happen. The sequence of events seemed to go like this. The white player (who I shall call Scully) played a move, putting his opponent (who I shall call Mulder) in check. Now Mulder did not notice, and played a move putting Scully in check. At this point Scully simply stopped the clock but said nothing. Mulder, who thought he was winning, extended his hand, believing Scully was resigning. Scully accepted the offered hand, believing that Mulder realised he'd played an illegal move and was himself resigning. (NB At Street Chess we play second illegal move loses). Now I'm not sure which of the players realised something had gone awry, but at this point I was called over. Further confusion ensued as Scully was worried he'd done something illegal in stopping the clock (no, but he should have told his opponent why), and then decided to resign. Realising what had happened, I gave Scully 2 extra minutes, told him he wasn't to resign yet, and to continue the game. 

Unfortunately I had to return to the same game a few minutes later when another issue arose. By this stage both players were short of time, so after Scully moved, Mulder replied instantly (and before Scully had pressed his clock). Scully then pressed his clock, completing his last move ( which I encourage under these circumstances), and Mulder then pressed the clock (without moving of course), to complete his move. However this confused Scully, who thought that Mulder had not played a move (even though he witnessed it). About half way through me going over this issue with the players (and in the midst of a gathering crowd), Mulder offered Scully a draw, and rather than listen to me lecturing them, shook hands and split the point.