Chess for any number of players

I found a Variant that generalizes chess to any number of armies, with each player getting half of a regular chess board, meaning an 8 by 4 board.  To make a move a player puts two of the half boards together.  One interesting feature of this variant, is that with more than 2 players bishops, and other normally color bound pieces, stop being color bound as a bishop, or other normally color bound piece, can go from half board A to half board B, half board B to half board C, and then from half board C to half board A to end up on a different color than it started on.  I suspect this would make normally color bound pieces become much more powerful in this variant.

1. Not feasible on chess.com

2. How exactly does this un-color bind pieces?

Well in this chess variant any pair of half chess boards has mirror symmetry from the initial position.

Let's say that we start with something like this, with all the kings facing each other and none of the kings moving during the demonstration.  Remember that the red king is on the dark red square as it will be important later.

Now let's move the bishop along the long diagonal to the other side of the red blue pair of half boards to get this

Now let's switch to view the green blue pair of half boards to get this view of the second position

Now let's move the bishop along the long diagonal to get it to the green half board to get this position

Now let's change the view to the green red pair of half boards

Notice that the red king is still on a dark red square, and the green king is on a dark green square, but the pair of half boards still has the red and green kings facing each other, so light green squares connect to dark red squares along diagonals, so when we move the bishop back to its home territory we get a position like this

with the bishop now on a different color than it started on if you treat dark and light red as different colors.