When both players use the Fabulous FIDEs, we have a game that I usually call "FIDE chess", but which others call "western shogi" or "western xiangqi"; this is the game that Morphy and Fischer played, and it's a darned good game and I love it, but since Chess was originally devised as a simulation of a battle in a war, it seems wrong for both sides to use the same army.
When has there ever been a battle, or even a sporting event, where both sides had exactly the same army or team? When the Yankees played the Dodgers, their catchers and shortstops had different abilities, and when Scipio defeated Hannibal at Zama, only one side used elephants.
For years, I researched the Values of Chess pieces in order to be able to devise good games of Chess with different armies.
I wanted to be able to let the players choose their pieces from a list of pieces and values, the way war games work with "buy points", or the way "fantasy leagues" work; I found that the values aren't sufficiently precise, and that the team as a whole must be considered. Some pieces work together well, and choosing them in combination adds to the value of the army as a whole; other pieces do not, and an army made up of them will be weaker on the board than it would seem to be "on paper".
Because of these problems, it seemed best to have a whole army designed by somebody who understands the problem of values -- me, of course!
As a byproduct of this research, a few games of "Chess [Variants] with somewhat different armies" were developed. However, although games such as Way of the Knight or Different Augmented Knights are good games, they are not nearly as perfect as a game where one side can use the traditional FIDE-chess pieces and the other side can use a completely different army (except for Kings and Pawns, which must stay the same).
The first such complete new army was The Colorbound Clobberers. In order to have names for the armies, I introduced the sports-team metaphor which causes some confusion because it is more natural to call them "armies" than "teams". Besides which, there have been armies with names, so the whole team thing was unnecessary to begin with.
I had always thought of "Chess with Different Armies" as a "Chess Variant", but when Vlad Roytman was good enough to play a series of games against me, I was shocked to discover that "Chess with Different Armies" is Chess, pure and simple!
Yes, this is Chess, "real Chess". It still feels like Chess, the games are exactly as long and exactly as difficult as Chess, the opening, endgames, and middlegames are just like Chess. Development, centralization, overprotection, initiative, weak-square complexes, they are all here. This is Chess. Heavy-duty, serious Chess.
But it's also all so different, of course. You can throw away the opening variations you memorized (but keep the general principles you learned from them). When you're trying to win a tough endgame in which you have a piece that moves like a Rook-plus-Knight, you can't look it up in Cheron or Averbakh (but many of the methods you learned there are still useful).
Once the Clobberers were devised and tested, it became a bit easier to invent other new armies. The two reasons why it became easier are that I got a better feel for how accurate the different measurements of value were, and that I learned it is not necessary to be too precise in matching the values of the pieces (my new rule of thumb: when each individual piece is within 10%, and the over/under is balanced, the army as a whole probably works).
After that, I finally came up with Buypoint Chess, but Buypoint is a Chess Variant, not a game of Chess.
Only when you play with the few teams in this section is it really "Chess with Different Armies"; but of course, I can also offer you some Chess Variants with Different Armies.