The state of living in a tribe is tribalism. This word is also used to describe situations where people are overly loyal to their own group. For example, a newspaper editorial might complain about tribalism in American politics.
This word was exclusively used to describe aspects of living in a traditional tribe until the mid-20th century, when it came to have a more derogatory meaning. Today, tribalism is often seen as putting one's own group above every other consideration, including kindness or justice. Tribalism can lead to bigotry and racism and, when taken to extremes, even war.