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Indeed, this is what I meant. That the Fairy-Max derivative that plays Alice Chess does it this way is completely accidental; initially I had misunderstood the rules to be such that moves had to be pseudo-legal on the board they were played, and that you only had to worry about your King being capturable after the transfer. This made it virtually impossible to checkmate a King, though, as he keeps fleeing to the other board.

So I patched it for not being allowed to pass through check. Internally it uses the single-board representation, and a flag in the piece code to indicate on which board it is. That flag is toggled every time you move the piece. When a move attempts to go to a square that is occupied by a piece marked as being on the other board, that square counts as empty and the non-capture move rights for that square are suppressed. Those were basically the only changes that were needed to make it play Alice Chess, except for this passing-through-check rule. So I later patched the code to make the test for royalty of the victim before it checks on which board it is, and if it is on the other board, abort due to King capture only if that square is also the to-square of the previous move (which was passed from parent to daughter node anyway for implementing Berolina e.p. captures).

This catches passing through check on any King move, but there is no test for King capture between making the move N transfering to the other board. A move going over that square would either skip over the square on the board you just left, but would block any moves over that square on the board you just arrived on. To implement the other rule interpretation, it would have to continue moves through the to-square of the previous move, solely for the purpose if there was a King behind it. It does not do that.

Note that the "legal if strictly legal on the board it is played" interpretation is flawed anyway, because it would allow you to legally put your own King in check: white Ke1, Pe2, black Qe8 (all on same board) would have e2-e4 legal, because before you transfer e4 to e4*, your King is not yet in check.