Anti-Relay Chess is a game concept that can be implemented in many different games, in combination with Relay Chess or alone. The number of possible implementations of the anti-relay concept is quite large. I will offer some definitions and a set of rules to attempt to cut the number down to size, while leaving game designers who choose to use them ample choices.
The relay concept is well known and its most common mode of operation is seen in Knight Relay Chess, in which a piece defended by a Knight gains the temporary ability to move as a Knight. Generalizing, a relay is when a piece gains a movement capability it normally does not have due to its position relative to another piece.
The anti-relay concept follows logically: an anti-relay is when a piece loses a movement capability it normally does have due to its position relative to another piece.
Natural move: A piece's natural moves are those move-types which a piece has without regard to relays and anti-relays. A Knight's natural move is the Knight move, a Queen's natural moves are the Rook move and the Bishop move.
Relay: A piece gains a move type which is not a natural move of that piece.
Anti-relay: A piece loses a natural move type.
See: A piece sees another piece if it could capture an enemy piece on its square using one of its natural moves. A piece sees another piece even it it could not actually move to the square due to an anti-relay, pinning, etc. The move type by which the one piece sees the other is the move type that will be given by relay or taken away by anti-relay.
Direct: Relays are always direct: the move is added to the piece that is seen. In a direct anti-relay the loss of move applies to the piece that is seen. This could also be called a Basilisk effect.
Indirect: An anti-relay where the loss of move applies to the piece which sees another piece. This could also be called a Medusa effect.
Normal: An indirect anti-relay which applies only when the two pieces involved share the seeing move.
Converse: An indirect anti-relay which applies only when the two pieces involved do not share the seeing move.
Total: An indirect anti-relay which applies in both the normal and converse cases.
Friendly: An effect applies when a piece sees/is seen by a friendly piece.
Hostile: An effect applies when a piece sees/is seen by an enemy piece.
Bilateral: An effect applies when a piece sees/is seen by a piece of either side.
Kings and pawns do not participate in relays and anti-relays. They neither gain nor lose moves, nor do they give or take away moves.
Relays and anti-relays do not cancel. A Queen which loses its Bishop move cannot regain it by a relay from a Bishop, a Queen which gains a Knight move cannot lose it by an anti-relay from a Knight.
Relays and anti-relays do not propagate. A Rook which gains a Bishop move by relay can't use that Bishop move to give a Knight a Bishop move.
Check, mate, and stalemate are calculated by taking relays and anti-relays into account. A King may move to a1 even though an enemy Bishop is on b2, if that Bishop cannot move due to an anti-relay. Any move by the Bishop's owner that removes the anti-relay is check, any move by the King's owner that removes the anti-relay is illegal.
Anti-relays are quite useful to provide a counterbalance when adding relays to a game. Relay alone make the pieces very powerful and make the game shorter and more tactical. This can be fun, but if in a particular game the effect goes too far, anti-relays can bring it back into balance.
Anti-relays can be used alone to slow down fast, highly tactical games such as Tripunch Chess.
In general, anti-relays have a levelling effect: strong pieces with many move types can lose moves easily. This is particularly true in combination with relays, as weak pieces can gain moves easily.
A game may be described by prefixing the types of relays and anti-relays to the name of the base game. You could have for example Hostile Direct Anti-Relay Chess. This would be FIDE Chess, with the provision that a Knight attacked by an enemy Knight loses its Knight move.
Another example: Friendly Relay Hostile Normal Indirect Anti-Relay Gothic Chess. This would be Gothic Chess where a piece defended by a friendly Knight gains a Knight's move, and a piece attacking an enemy Knight, Chancellor, or Cardinal via a Knight move loses its Knight move. In this game, a piece can't capture a piece using a shared move type.
A third example is my current favorite in this genre: Friendly Relay Hostile Converse Indirect Anti-Relay Tutti-Fruitti Chess. This is Tutti-Fruitti Chess where a piece attacking an enemy Rook, Bishop, or Queen via a Knight move loses its Knight move. In this game a piece can't capture a piece without using a shared move type or a relay power.
Naming can be simplified by some conventional defaults:
Relays are assumed to be friendly.
Anti-relays are assumed to be hostile and direct.
Indirect anti-relays are assumed to be normal.
If a game uses converse or total indirect anti-relays, the word indirect may be omitted.
So the three examples become Anti-Relay Chess, Relay Indirect Anti-Relay Gothic Chess, and Relay Converse Anti-Relay Tutti-Fruitti Chess.
As an exercise, describe the rules of Hostile Relay Friendly Anti-Relay Grand Chess. I wonder how this would play.