What is the default Extra Administrative Realms — Players game rule in CK3?

No Special Changes: Players will only be Administrative if they select a Realm that already has the appropriate Government. (patch 1.20.1.2)

Does changing the Extra Administrative Realms — Players rule disable achievements?

No: none of its 3 settings carries the flag blocks_achievements in patch 1.20.1.2.

Game rule · Historicity · Game Modes

Extra Administrative Realms — Players

No Special Changes · default
Always
With August Trait

Extra Administrative Realms — Players is a game rule in Crusader Kings III (Historicity and Game Modes) with 3 settings: No Special Changes (the default), Always and With August Trait. None of its settings disables achievements. Patch 1.20.1.2, from game/common/game_rules/00_game_rules.txt.

This section as JSON: /raw/game-rules.json

Quick answers (2)

What is the default Extra Administrative Realms — Players game rule in CK3?: No Special Changes: Players will only be Administrative if they select a Realm that already has the appropriate Government. (patch 1.20.1.2)
Does changing the Extra Administrative Realms — Players rule disable achievements?: No: none of its 3 settings carries the flag blocks_achievements in patch 1.20.1.2.

3 settings

No Special Changes
Default
Players will only be Administrative if they select a Realm that already has the appropriate Government.
The script checks it once, in 1 place
On actions (the game's own hooks) 1
- on_game_start_after_lobby
Every has_game_rule = historicity_extra_admin_player_wherever_gameplay_not_administrative in game/common and game/events, by the top-level object it sits in.
historicity_extra_admin_player_wherever_gameplay_not_administrative
Always
Players that begin with at least a Kingdom-Tier Realm will automatically convert it to an Administrative Government on game start.
The script checks it once, in 1 place
On actions (the game's own hooks) 1
- on_game_start_after_lobby
Every has_game_rule = historicity_extra_admin_player_wherever_alt_historic_always in game/common and game/events, by the top-level object it sits in.
historicity_extra_admin_player_wherever_alt_historic_always
With August Trait
Players that have the August Trait & begin with at least a Kingdom-Tier Realm will automatically convert it to an Administrative Government on game start (intended for asynchronous Governments in multiplayer).
The script checks it once, in 1 place
On actions (the game's own hooks) 1
- on_game_start_after_lobby
Every has_game_rule = historicity_extra_admin_player_wherever_alt_historic_with_august_trait in game/common and game/events, by the top-level object it sits in.
historicity_extra_admin_player_wherever_alt_historic_with_august_trait

Source: game/common/game_rules/00_game_rules.txt line 1499 (key historicity_extra_admin_player_wherever), modifiers from game/common/modifiers, names from the English localisation; Crusader Kings III 1.20.1.2.