In Duro v. Reina, 495 U.S. 676, 685, 688 (1990), the U.S. Supreme Court held that, except as provided by Congress, tribes lack criminal jurisdiction over nonmember Indians. In other words, an Indian tribe cannot prosecute Indians who are not members of that tribe, except as authorized by Congress.
In response to this ruling, Congress amended the Indian Self Governance section of the Indian Civil Rights Act to state that Indians have the right to prosecute anyone who commits a crime on that tribe's territory, regardless of Tribal Affiliation.