The Imperial Japanese Army and the Navy had a rather notorious interservice rivalry where the basic political factions in Japan (after the Taisho democracy had been done away with) were the Army and the Navy. Both carried out assassinations against each other in the early 30s, and the Army advocated for securing resources in Siberia ("Strike north faction"- the brief war with the USSR ended that quickly) while the Navy advocated for securing resources in the Dutch East Indies and elsewhere ("Strike south"- this gained prominence after the failed campaign against the USSR and the Army lost prestige).
Eventually, the rivalry had gotten to the point where the Army had its own aircraft carriers and submarines, and the Navy had its own infantry and paratrooper forces.
The challenge here is to have this rivalry (and assasinations) increase to the point where a civil war occurs between the two branches. Ideally anywhere from the 1920s to the 1940s.