Rules for King restoration
Here, a set of rules is given where the Time Thief is so powerful that he can restore the allied King.
In the basic set of rules, a Time Thief cannot restore the allied King. In this advanced set of rules, a Time Thief can in fact restore an allied King.
If a player has a Time Thief in his army, then the FIDE laws for check, mate and stalemate do not apply to this player. It is legal for this player to expose the own King to an attack.
This player loses the game, when his King is captured, and he fails to restore his King with his Time Thief on the very next move. This player even loses the game, when he answers with capturing the opponent's King instead of restoring his own King.
For a player without a Time Thief: getting out of check must be done before capturing the opponent's King.
A player with a Time Thief survives and can play on, as long as he is able to restore his King immediately after losing his King.
As soon as a player loses his Time Thief and still has his King (*), then the FIDE laws for check, mate and stalemate immediately apply to this player.
(*)"... and still has his King" has been added, as to anticipate that people combine the advanced Time Thief rules with other pieces that have double capture abilities.
Examples
White to move: ixj7 is a legal move. Nxe1 is answered with TTxf3 and White still lives.
White to move: ixj7 is a legal move. But not a good move. After Nxe1, TTxf3 is possible, but then after Bxe1 White cannot restore his King and loses.
Black to move: Black has no Time Thief in his army. He must get out of check, and Nxe1 is not a legal move for Black.
White to move here. Let's assume that White plays jxi8. TTxj7 is the only move for Black, or else Black loses. Black might be tempted to play Nxe1 instead, but that loses instantly. After losing his King, Black must restore his King on the very next move.
White to move: White is in big trouble and whatever move he can do is losing.
Let's see:
1. Ne3 Rxi1 with mate, as White's Time Thief is captured and FIDE rules apply to White now, and it's badly mate.
1. Ng1 Bxj1 King cannot be restored and white loses.
1. Kxj2 Rxj2 King cannot be restored and white loses.
1. TTg1 Qxj1 King cannot be restored and white loses.
In previous position, it was Black to move, and Black decided to take the Knight on d1.
Again, no matter what White does, he'll lose the King and - with that - the game.
Now in this diagram:
If it was White to move: stalemate.
If it was Black to move: Ke4# is legal for Black and wins.
Players with the Time Thief can choose to play with or without this advanced set of rules with King Restoration, as long as they discuss and agree upfront.
This is an enhanced version of a previous explanation:
https://www.chess.com/clubs/forum/view/the-time-thief-explained
The Time Thief moves like a Queen.
The Time Thief can be captured by other pieces. But the Time Thief itself cannot capture other pieces in a normal way.
However, the Time Thief has a special way of postponed capture:
If an enemy piece is in reach of the Time Thief, and does a move, the Time Thief can as response revert the entire move and then capture the piece.
In this diagram, the black Rook is in reach of the white Time Thief. Suppose Black plays Rf2.
As response, the white Time Thief can revert Black's previous move. So it would move the black Rook back from f2 to f7. And then it can capture the black Rook on f7. As a result this would be the position:
The Time Thief must do this capture on the very next turn, or the right to do so is lost!
The Time Thief cannot revert a move after being captured itself:
After Rxh3, the Time Thief is off the board, and has no powers to revert that move and capture the Rook.
If the Time Thief reverts a move, it reverts all aspects and everything what happened with that move. It really turns back time.
In this diagram, Black might play Rxi7. The Time Thief can revert that move and capture the Rook on f7, but that also means that the Queen comes back on the board on i7 : the Queen is restored by the Time Thief.
After Rxi7 TTxf7:
In the following diagram, Black has just played pawn f7-f5:
Before f7-f5 was played, the Rook was in reach of the Time Thief. but the Rook did not make any move. So the Time Thief cannot capture the Rook.
The pawn did move, but did not come from a square in reach of the Time Thief. So the Time Thief can also not capture the pawn.
In the following diagram, the White King is in check.
White might consider to play ixj7, thinking that his King is indirectly protected by the restoring powers of the allied Time Thief. This is not allowed. A Time Thief cannot restore his own allied King. In this position, White must get out of check.
A King in reach of an enemy Time Thief, is not yet in check.
White to move here. With any move by the White King, White would put himself in check, because it allows that the black Time Thief will capture the white King. This position would be Stalemate, as White has clearly no legal moves left.
Note that in the same position with black to move, Ke4+ would be an illegal move for Black. Although the White King seems to be frozen by the Black Time Thief, he still gives check, if the Black King would approach to an adjacent square.
Fairy piece interactions
Time Thief and Spirit
Scenario: a possessed piece is in reach of a Time Thief and the Spirit leaves that piece.
Additional rule 1: If the Time Thief is unpossessed then the lone Spirit is invulnerable for the Time Thief. The Time Thief can't take.
Additional rule 2: If the Time Thief is possessed then the Time Thief can capture the Spirit. The piece that was possessed by the Spirit will as a consequence also be captured. This is a circumstantial double capture and is allowed. If the possessed piece was a Dwarf then the circumstantial double capture is still allowed whether or not the possessed Time Thief had an allied co-attacker.
Other scenario's with Spirit and Time Thief are already covered by the existent rules.