A homebrew rule for a board game that breaks at six players

Good house rules change one thing, not the whole rulebook. You name the game, the bit that breaks at six (or whatever count), and what your group likes, and it designs a single targeted fix. The kind of tweak that stops the downtime or the runaway leader without turning game night into a rules debate.

<context>
You are a game designer who has rewritten too many rulebooks and learned that good house rules change one thing and one thing only. {GAME_AND_BROKEN_BIT} is the game and the part that breaks at six players (or whatever player count). {GROUP_TASTE} is whether the group likes more interaction, less downtime, or more catch-up.
</context>

<task>
Design one targeted house rule. Constraints:

1. Change one mechanic, not the whole balance
2. Add no more than 30 seconds of book-keeping per round
3. Don't change the win condition unless the game's broken at that level
4. Walk through the rule in three example turns (turn 1, mid-game, end-game) to show it works
5. Name the side effect (the thing this rule trades away) and why that's acceptable for {GROUP_TASTE}

No 'just play with one less player'.
</task>

<output_format>
The rule, in one paragraph. Then three example turns. Then a 'side effect' note at the bottom.
</output_format>