Avignonisme
The most common depicition of a "French Cross", the primary symbol of Avigninisme
[...] The Great European Revolutions was a great social upset in many ways, rapturing the foundations of the connection between religion and politics with the last vestiges of the divine right of kings being swiftly followed by a new era of secular, liberal and rational thought. Religion was in many ways still seen as a cornerstone of European culture, but in addition to rupturing the spiritual basis of many church traditions, the revolutions also caused a large upheaval in the logistics and organization of many religious organizations at the time, perhaps none as prominent as that of the Catholic church. In the aftermath of Italian unification and Pope Pius IX's very hesitant endorsement of the new Italian Republic, many catholic traditionalists saw his authority as nullified on the basis that he was now a participant in a temporal nation's affairs and had thus forfeited his spiritual right to lead even as the Pope himself publicly declared his political neutrality from St. Peters Basilica.
Of the many "National Catholic" churches that sprang up in it's aftermath, none would perhaps be more geopolitically influential than French Catholicism, often called "Avignonisme" after the movement's traditional seat at Avignon. Following the wave of revolutions and in particular the restoration of the French monarchy the French cardinal of Bourges Jacques-Marie-Antoine-Célestin Dupont declared Pius' authority nullified and convened a "special council" of prominent conservative catholics that elected him the new Pope under the name of Gregory XVII (a deliberate shot at Pius, as Gregory XVI had been his predecessor). In an irony most likely not lost to those involved, Gregory XVII chose his former home of Avignon to host the French catholic church.
This church could well have remained an obscure splinter like the Irish papacy were it not for the Monarchist coup and the subsequent inviation by French King Henri V to perform the official coronation. The French catholic church was subsequently declared the offical church of all France and was granted wide-ranging authority convince the populace of its legitimacy. Althought a not insignificant amount of conservative Frenchmen were genuinely swayed by the church's theological arguments, a far greater number had to face the threat of force before promptly professing their undying loyalty. The church found a suprising ally in the form of several metropolitan churches that unlike many loyalist church officials saw the birth of the Avigon church as either a natural step in the evolution of the French nation, a vital step to protect the good people of France from the vile temptations of a foreign Pope or some combination of both.
The church saw its largest support during the First Global War as a civilian organization helping wounded, refugees and the poor both during and after the war but lost a large amount of institutional influence following the defeat of France and the subsequent abdication of its Monarch, but was nonetheless spared (to the annoyance of some italians) any threat of dissolution or dispandment. Despite this it lived on and had a brief revival during the Labryist regime in an inferior position as somewhat of a glorified propaganda organization in contrast to the more equal position it held during the royalist period. Avignonisme is today mostly a spent force, but still maintains a sizeable following in souther France compared to its international counterparts. The current Pope Julius IV still maintains the claim of being the true leader of the Catholic faith, but has still adopted a more conciliatory tone towards the Roman Papacy than his forebears.