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Kevin:

I do not know what would happen if the case referred to is met. If you help me explain the answer, I gladly include it in the rules.

Hi Carlos

One solution is:

1) You must try to get any of your kings that is in check out of check (as per FIDE rules).

2) If you cannot get all of your kings that are in check out of check, then you lose. Thus it's possible to simply check both kings at once, in a way that neither is checkmated, and still the game comes to an end if both kings cannot get out of check.

This solution (which I'd prefer, though perhaps it's not in the spirit of your variant) would have the advantage that it has been used in other games with more than one royal king per side. However, if as your preset states at the moment you must (literally) capture both kings to win, and you want to keep that condition, then I think you'd have to forget about the FIDE rule compelling a player to try to get any of his kings out of check, and allow a king to be (literally) captured sometimes (in a way your current castling rules do this, as castling through squares which would put a king in check is allowed). With that condition, I suppose it's possible for a player to 'checkmate' both of his opponent's kings, yet the following move being only able to literally capture one of them (e.g. if a single piece is participating in the checkmate of both kings, and/or one of the checkmated kings makes a run for it with a move that still leaves it in check, illegal if FIDE rules were still to be used regarding kings).

At the moment I cannot think of any other 'clean' solutions than these two, but maybe someone else can (or has), For what it's worth, in two player 4 army variants with more than one army per side, one of a player's army's kings can be 'checkmated', with the condition that it cannot be captured, but the rest of that army is immobilized unless the checkmate is relieved somehow; in such a case the immobilized army's pieces may or may not be allowed to be captured, or the mating player even takes control of that single army, too, from then on. I suppose in a 2 single armies variant with 2 kings per side, it could be stipulated that there's some sort of a penalty for having one of the kings checkmated (with the loss of a game for having both checkmated), but then there's still the awkward matter of what to do if both kings are merely in check, but both cannot get out of it at the same time.

Reference to FIDE rules is moot, as those rules could never create a position where one player has more than one King. It all depends on how you generalize the rule that "it is not legal to expose your King" to capture. With multiple Kings this could be done as "it is not legal to expose any of your Kings to capture", (absolute royalty), or as "it is not legal to expose your only King to capture" (extinction royalty). The concept of 'check' would be defined as a situation where it would not be legal to pass your turn when turn passing would have been a legal move in general. So with extinction royaly, if one of your two Kings is attacked, this is not considered check, and you are under no obligation to resolve it. You can just allow it to be captured, no harm done.

In Spartan Chess the rule is somewhat intermediate: "it is not legal to expose all your Kings simultaneously to capture". So there you lose when both your Kings are 'checkmated', although the latter is just a figure of speech, as it is actually the player that is checkmated when he cannot get out of check, while check is a situation where both Kings are attacked. 'Checkmating' one of the Kings then merely indicates that you can force capture of that King on the next move, which the player owning it might not mind at all if that King was protected, so that he can recapture.

In Spartan Chess a second King is worth 4.5, i.e. slightly less than a Rook, so protecting it is a good defense against capture by Rook or Queen. (This depends of course a bit on how save your remaining King is.) With two absolute royals, the value of the second royal is about 0, even in absense of Queens, i.e. it is as much a liability as an asset. In Pawn endings it is a great asset, however, and two Kings versus one would win such endings easily. With the Queens still on the board, it would probably have a negative value. (But the value is only good for estimating who is winning, as you would not be able to trade the King under those rules.)