Schism
Etymology
From Greek "skhisma" (a split, a cleft), from "skhizein" (to split) — the same root behind "schizo-." It came into English through church history, above all the Great Schism that divided Eastern and Western Christianity, and kept that flavor of a deep, institutional cleaving.
noun · /SIZ-uhm/
A formal split or division within a group, especially a church or movement.
What it names
A rupture that becomes official — when a faction does not merely disagree but breaks away into a separate body.
Example
A dispute over strategy hardened into a schism, and half the party left to found their own.
Related idea
A rift can be a private, informal falling-out; a schism is a formal, often institutional split, especially within a church or movement.