No, it's not historically accurate. What the game engine is trying to do - hamfistedly - is simulate political conflict within the Republican coalition, which while not entirely peaceful was certainly nothing resembling a three-way civil war. This conflict did boil over into actual violence in one notable instance in May 1937, when government forces attempted to seize key public utilities in Barcelona that were occupied and controlled at the time by anarchist and other leftist militias. When these militias attempted to resist, this led to confused street fighting in Barcelona for several days before a settlement was reached.

Importantly, while anarchist forces were involved in the initial street fighting, the major anarchist leaders and organisations ultimately did not break with the Republican government. Rather, the target of official repression (egged on by Spanish communists in particular) became the Partido Obrero de Unificación Marxista (POUM), a relatively small revolutionary leftist party who were vocal opponents of Stalinism. They were scapegoated for the street fighting, and portrayed as secret allies of the military rebellion, seeking to launch an uprising in the rear to aid Franco's latest offensive. I go into the wider political context of this conflict in this older answer, but the result was the wholesale repression of the POUM in the months that followed, with many members imprisoned and some key leaders murdered. While now commonly seen as a blight on the wartime record of the Republican system, it's important to note that the relative scale and importance of these events is often exaggerated in popular representations of the conflict because George Orwell happened to serve with the POUM militias, and his experiences of the fighting and subsequent repression of the POUM played a central role in his famous account of the conflict, Homage to Catalonia.

While the POUM was suppressed, this was more in the nature of internal policing rather than open conflict as portrayed in HOI4, precisely because the POUM simply wasn't that big and couldn't exactly raise entire armies. The government was willing to negotiate with anarchists (albeit from a position of strength) precisely because anarchist groups represented a much larger number of people, and a full-scale anarchist uprising against the Republic (per HOI4) would have had dire consequences for the war effort. For their part, anarchist leaders were generally willing to compromise than prolong an internal conflict that would play into Franco's hands. While Spanish liberals, socialists, communists and anarchists might all have very different political ideas about the conflict and its goals, they all did share a commitment to defeating what they saw as a fascist rebellion. So while internal Republican politics were never exactly easy or straightforward, the broad anti-fascist coalition did manage to hold together until the very final weeks of the war, and nothing like a three-way war ever emerged.