In Phantasmagoria of Dim. Dream it is possible to die at the same time as your opponent, but since a "Draw" doesn't actually exist, one of the players would "Win" and the other one would "Lose".
The two situations have been showcased by ZillionBrain in his two videos: youtube.com/watch?v=Wsi8b2f2lRQ and youtube.com/watch?v=uPStk8qy3lw. The gifs in this post are from these two videos.
When the player "wins", no status screen is shown at the end of the stage and you don't advance to the next stage. Instead, you lose a life and have to replay the previous stage. If you are on your last life and this happens, you will get a Game Over, as described by user KirbyComment in a comment to the video.
If the opponent "wins", the status screen is shown on their side at the end of the stage (granting the winning points to the opponent) and the game proceeds to the next stage. The player doesn't actually lose any life.
If you look at the behaviour of these two "wins" it's like the opposite is happening. Usually, when you lose there is no status screen, you lose a life and you redo the stage. But when you win there is a status screen, you don't lose any life and you proceed to the next stage.
