Emergent narrative
Also said as Embedded narrative (its opposite), Player stories, Systemic storytelling
Embedded narrative is written in advance and waits to be found, as a cutscene or an audio log. Emergent narrative comes out of the rules running, so players end up telling stories nobody wrote. Henry Jenkins set out both in a 2004 essay.
Embedded story gives you control and costs words. Emergent story costs systems instead, and the player owns it, but you cannot promise it will land or be any good.
Say it
“I want emergent stories from the village: give each person a grudge list and a hunger value, let theft raise the grudge, and print what happens as a plain log. Don't script any of it.”
Where you’ve seen it
Middle-earth: Shadow of Mordor's Nemesis system lets an orc who kills you get promoted, remember the fight and carry the scars when you meet again. BioShock's audio diaries are the embedded kind.
Not to be confused with branching narrative
A branch is still written by someone, in advance. An emergent story was never written at all: only the rules that produced it were.