Hi.
My playgroup does 3HG (we are 6 = 2 teams)
Say we face like this:
A + B + C
-vs-
X + Y + Z
And One of the XYZ guys wins the die roll are play first. Regardless of being clockwise (which is the default) or counter clockwise (which is not) team ABC gets turn 2.
Now the question:
What's the order of choosing?
If Z plays it, would Y be next, and then X, and then A -> B -> C?
(I ask this to choose wisely where I'll sit next time).
In any game with the shared team turns option, including Two-Headed Giant and Three-Headed Giant (C.R. 810.2, 810.11), whenever something has multiple teams do something at once, first the active team (the team whose turn it is) makes all choices required, then each nonactive team in turn order (C.R. 805.6). In general, this order can be decided by the team's players in each case (the relevant part of C.R. 805.6 says: "... first each player on the active team makes any choices required in whatever order they like", that is, in whatever order the team likes, "then the players on each nonactive team in turn order do the same"). If the players on that team "can't agree on [that] choice ..., the primary player" (that is, "the player seated in the rightmost seat from that team's perspective") "makes that choice" (C.R. 805.2).
If we follow the order suggested in C.R. 805.6, then in a Three-Headed Giant game, if Team A/B/C is the active team, and Thieves' Auction (controlled by any player on that team) resolves, "each player" in that team, in whatever order the team likes, but "[s]tarting with" the controller of Thieves' Auction, "chooses one of the exiled cards and puts it onto the battlefield under their control", then each player on the nonactive team does the same, and this process repeats "until all cards exiled this way have been chosen" (C.R. 108.1, 109.5, 101.1).
Note:
It appears that the cited portion of C.R. 805.6 has become ambiguous since the transition in Dominaria to the singular "they" in the rules. Notably, even before Dominaria, C.R. 805.6 (as was in effect in Rivals of Ixalan) still said "first each player on the active team makes any choices required in whatever order they like" (that is, whatever order the team likes, not whatever order the player likes), and this has remained unchanged with Dominaria and beyond, at the time of this writing. Thus, in general, the team decides the order in which its players perform actions if more than one of them has to perform actions or make choices at the same time.
EDIT: Edited after comment 3 was posted.
EDIT (Oct. 9): Edited, including for correctness.
My playgroup does 3HG (we are 6 = 2 teams)
Say we face like this:
A + B + C
-vs-
X + Y + Z
And One of the XYZ guys wins the die roll are play first. Regardless of being clockwise (which is the default) or counter clockwise (which is not) team ABC gets turn 2.
Now the question:
Later on Thieves' Auction is played.
What's the order of choosing?
If Z plays it, would Y be next, and then X, and then A -> B -> C?
(I ask this to choose wisely where I'll sit next time).
If we follow the order suggested in C.R. 805.6, then in a Three-Headed Giant game, if Team A/B/C is the active team, and Thieves' Auction (controlled by any player on that team) resolves, "each player" in that team, in whatever order the team likes, but "[s]tarting with" the controller of Thieves' Auction, "chooses one of the exiled cards and puts it onto the battlefield under their control", then each player on the nonactive team does the same, and this process repeats "until all cards exiled this way have been chosen" (C.R. 108.1, 109.5, 101.1).
Note:
It appears that the cited portion of C.R. 805.6 has become ambiguous since the transition in Dominaria to the singular "they" in the rules. Notably, even before Dominaria, C.R. 805.6 (as was in effect in Rivals of Ixalan) still said "first each player on the active team makes any choices required in whatever order they like" (that is, whatever order the team likes, not whatever order the player likes), and this has remained unchanged with Dominaria and beyond, at the time of this writing. Thus, in general, the team decides the order in which its players perform actions if more than one of them has to perform actions or make choices at the same time.
EDIT: Edited after comment 3 was posted.
EDIT (Oct. 9): Edited, including for correctness.