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If the player becomes an illegal target, the target permanents are tapped, but the player doesn't skip their untap step.
If more than one effect instructs a player to skip their next untap step, such as if Yosei dies more than once in a turn, that player skips that many untap steps. This is different from effects that state that something doesn't untap during an untap step.
If a permanent won't untap during its controller's next untap step, that effect waits to apply until its controller has an untap step that isn't skipped.
If any of the targets of Yosei's ability are illegal, they won't be tapped.
If all of the permanents targeted by Yosei's ability become illegal targets, nothing becomes tapped, but the target player still skips their untap step.
