You wake up and find yourself in a vast expanse of emptiness. As you look off into the distance you see a faint speck contrasting with the pale white expanse. Lacking anything better to do, you decide to head for it.
As you start walking, you feel that the distance between you is closing unusually quickly. It does not take you long to realize that the speck is moving towards you to. Before long, you begin to distinguish details. It is not a speck, but actually a person. You call out, but no response. Perhaps they did not hear you. As you get closer you decide to make another attempt. As you call you seem to hear something coming from the other person, and quickly quiet down.
They do the same. After a few false starts you decide to just keep walking towards them, as they begin to walk towards you. As you draw closer you begin to distinguish details, this person looks rather familiar. Suddenly it dawns on you: they look just like you. You start, and they stutter as well.
"The only way to straighten this out is by talking" you think, so you both decide to close the distance. As you draw closer you attempt to start a conversation, but every time you say something they say something too. You stop to let them continue, and the halt as well. Finally, you are face to face. It is almost like looking in a mirror. You turn to look at a different angle, and they turn as well. You say something and they do too. You walk around to your left, and they walk around to your right.
It is then that you realize that you are perfectly synchronized. Every time you try to speak, they say the same thing. Everytime you scratch your head, they do as well. You begin to wonder, if you are so well synchronized, surely the same thoughts are running through your head. You try to figure a way to break this symmetry, any way that you can actually hold a conversation with your other self. One where you are both not cutting off and falling silent.
How do you do it. How do your break a perfect mirror without outside help?
Edit: The universe isn't necessarily deterministic, at least I feel that it doesn't require it. But otherwise for all intent and purposes the 2 you/mes are mirrored down to the quantum level.