Are there wars that saw three different groups all fighting one another?

(My answer on Quora)

Yes… there are many.

The classic example in the 20th century was the Sino-Japanese War (1937–1945). Both the Chinese Nationalists and Communists battled the invading Japanese while fighting each other at the same time. Contrary to popular mythology the efficacy of the truce between the two Chinese groups was indeed greatly exaggerated.

Another example was the fighting between the two Palestinian terrorist groups Fatah and the PFLP. Both were at war with Israel while maintaining a state of belligerence toward one another.

Rival Islamist groups often fight each other while having a common enemy in the Great Satan that is the US. Personalities and questions around doctrinal purity often feature here.

Hanoi dominated Vietnam and the Khmer Rouge in Cambodia were also hostile to each other in the 1970s regardless of the fact that they were opposed to the United States.

In the Old Soviet Union Trotskyites and Stalinists loathed one another often engaging in hostility elsewhere despite having an overlapping enemy in the Western capitalist democracies.

Going back to the 19th century, there was no love lost between far left anarchists and their more orthodox Marxist rivals. Each furnished a strong animosity towards the bourgeoisie governments in Europe.