It wasn't really a "war", more about Phil being fed up with the DiMeo administration and wanting to decapiatate it and put people incharge who would listen to him. Most mob wars are inner family: The 90s Colombo war, Merlino/Stanfa war, Scarfo/Riccobene, etc.

Something like this almost took place in 2000. The Genovese wanted to start a bookmaking operation in South Philadelphia and conspired with the Philly North NJ Capo, Pete "The Crumb" Caprio to kill Joseph "Uncle Joe" Ligambi (Boss), Steven "Handsome Stevie" Mazzone (Underboss), and George "Freckles" Borgesi (Consigliere)...which the the Philadelphia family's administration.