David and I played around with the idea of a simultaneous chess variant until we found one that seems balanced. Rules:

Both players secretly select a move.
1. If both can execute without conflict, both do.
2. If both can execute only in a specific order, they are executed in that order.
3. Otherwise, one move is discarded at random and the other executes.
You may not choose a move that would end with your king in check, given the current board position.
You win by capturing or checkmating the opponent's king.
In selecting a move you choose a maximum number of spaces, but a capture before reaching the maximum counts as a successful move.
You may select a move if it could be legal after some opponent move. If it ends up not being legal, it doesn't happen.
To capture en passant you must attempt the capture simultaneously.
If you are using a clock, it only counts the time between the move selections.

Consider this position:

Say White chooses f3-f7 and Black anticipates that and chooses e8-f7. Black is allowed to make this move because (2) does not apply and (5) does. If we resolved Black's move first then only White's move could happen, but if we resolve White's first then both can happen, so we get the latter:

First White takes the pawn at f7, then Black takes the queen at f7. Alternatively, Black could have selected f7-f6. In that case the only way for both to happen would have been for Black's moved to happen first, at which point (4) means White's queen captures at f6 instead of f7:

This is a bad situation for White, because pawns very powerful. Because a diagonal move only succeeds if it is a capture, (1.3) means Black selecting g7-f6 forces a 50-50 for the White queen. Whereas, if the queen were threatened by any other piece it could just move away (1.2).

In writing this up I found that there is already a Simultaneous Chess variant. That version has a system of "initiative" to handle conflict resolution, which seems like it would play quite differently.

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