I just a random thought. I couldn't produce one off the top of my head and it seemed like any instance of it would be interesting.
Here are past answers & discussions on the topic:
[19 Apr. 2013 | 15 c] Flaired Respondents | Ancient Jerusalem, Medieval England, Modern Latin America
[04 Apr. 2012 | 42 c] Chinese Civil War, Ming China
[07 Sept. 2012 | 82 c] Russian Civil War, Spanish Civil War, Mexican Revolution
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The battles of 1066 in Britain come to mind, though I cannot vouch for the relation between the Vikings north Nd the Normans in the south.
The Tai-Ping Rebellion had more than three. The British fought against the Imperials and then against the Tai-Ping. There was also a third Chinese faction that was hostile to both the imperials and the Tai-Ping.
The Angolan Civil War, which launched after Angola gained independence from Portugal, was fought between two major former resistance movements but also featured intervention from the regimes of Apartheid South Africa, Communist Cuba, and others who were interested in using that conflict as a proxy for wider struggles in the Cold War.
This was a complicated conflict in which not every faction was belligerent with every other faction. It was, however, complicated enough that you might be interested in reading up about it.