If an effect says “You may search your library . . . If you do, shuffle your library” or “You may search your library . . . then shuffle your library,” your opponents can’t choose to search, so they won’t shuffle.

If an effect says “Search your library . . . then shuffle your library,” your opponents shuffle their libraries even though they can’t search.

Effects that instruct your opponents to reveal or look at cards from the top of their libraries will still work. Only effects that use the word “search” are affected.

An “extra turn” means a turn created by a spell or ability. It doesn’t refer to a turn taken in a sanctioned tournament after the time limit for the round has expired.

If an opponent’s extra turn is created while Stranglehold is on the battlefield, but Stranglehold leaves the battlefield before that turn would begin, that opponent takes the extra turn as normal.