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In Europe, between 1378 and 1415 there were Two Popes at all times and eventually Three Popes!
Question- What do you call it when they have two Popes at the same time?
Answer- Well, you usually call it temporary. You call one of the Popes - 'the Pope' and you call the other one 'the Anti-Pope'.1
Question- What happens when a Pope and an Anti-Pope collide?
Answer- Contrary to popular belief they don't eliminate one another and release huge amounts of energy 2. Usually one of the Popes has got enough support, so that the other one begins to look like a crank almost immediately. Once the weaker Pope is deposed his claim on the Seat of Paul is described as 'anti'. As in all matters, the victor gets to label the loser.
Question- So how did we get Three Popes at the same time?
Answer: Well, I'm glad you asked.
It all started with the Babylonian Captivity3 which began in 1309. The Archbishop of Bordeaux had recently become Pope and he noticed that Italy of the day displayed the sort of political stability today associated with Sierra Leone. So he moved the Papacy to the fortress town of Avignon in Provence, France. It was secure, it was fief of a vassal of Rome (so technically was Roman, not French) and it was in Provence. The time the Papacy was in Avignon is referred to as the Babylonian Captivity by later critics who saw a parallel with the forced exodus of the Israelites to Babylon from ancient times.
But what was Rome without the Pope in the 14th Century? Not a lot decided the people and cardinals alike. In 1328-1330 they tried declaring Nicholas V the true Pope - but it turned out he was an Anti-Pope because he was the first to back down when the Popes faced off. Avignon remained the seat of the Papacy until March 1378 when the real, actual and un-anti Pope Gregory XI died whilst in Rome.
Fearing the wrath of the Roman crowd, the cardinals elected Urban VI - who was Italian, but thought to be sympathetic to Avignon's claim on the Pope's HQ. As it turned out Urban was sympathetic to absolutely no one. Within months he had ticked off all and sundry - and his election was declared null and void by a bunch of Avignon-loving cardinals who packed their bags and elected Robert of Geneva to be the new Pope (Clement VII) when they got back home to Provence.
Urban VI might have become another in a long line of Anti-Popes with reigns of less than a few months - but for the support he had from the Italian public, some Rome-loving cardinals, England and Flanders4. He decided to stay Pope and sit back in Rome until his death in 1389. His successor was Boniface IX (1389-1404) who even got Portugal supporting him, strengthening the claim for Rome as the Papal Seat.
Meanwhile in Avignon, the other Pope garnered the support of France, Scotland, Aragon, Castille and lots of little German states. Clem 7 wasn't backing down and as far as he was concerned Urban VI should be called 'the other Pope' by amateur historians writing for h2g2.com and not him. When Clem 7 died in 1394, Avignon elected a Spaniard the new Pope: Benedict XIII. 13 was not their lucky number. Benedict was such a mongrel that even France (who started the whole Captivity) declared there was no Pope (not two, not one, but none!) from 1398 to 1403 and after patching things up, again from 1408 to 1409.
Rome was running smoother - with Innocent VII dying after two years and Gregory XII replacing him. But the general Catholic theological community decided having two, one and no Popes all at the same time was looking a bit strange. In 1409 they met at Pisa and decided it would best for all concerned if they declared Greg XII and Ben XIII both null and void and elect a new Pope - let's call him Alexander V.
France decided there was a Pope after all - Alexander V (the Council of Pisa had been their idea after all). Ben XIII and Greg XII decided the Council was stupid and that when all was said and done - they were both the one, true pontiff (along with the third Pope). When Alexander V died in 1410 the two Pope situation briefly emerged - except a successor to Alex V was elected - this being Pope John XXIII5 So, back to Three Popes.
John XXIII lacked the support of several major nations which he'd need to get Ben and Greg to stand down. Then there was John's personality. Within five years he'd lost most of his support. With all the nations of Europe calling for a solution, John convened the Council of Constance6. The Council promptly deposed both Benedict XIII and John XXIII. The third Pope - Gregory XII of Rome - declared 'you cant fire me, 'coz I quit' or Latin words to those effect. This was good fortune for him - because there was a strong Catholic tenet of the time which held that 'no power could depose the true pope.'
Martin V was elected in 1415 as the Pope. That's all. Just Pope Martin7. Of course Benedict XIII refused to believe the comedy was over and even after he was carted off to a Spanish castle to live in exile for the rest of his days he kept sending out ex-communications against people he didn't like, signed Pope Benedict XIII8.
So who was the Anti-Pope? or Anti-Popes? Well, because Gregory XII stood down voluntarily and thus was not removed, he is considered to be the Real, Actual, True, One Pope up to 1414. Thus his line - the Urban line or the Roman line - is now described as the Popes. Everyone else - all four of them - are Anti-Popes. Martin V returned to Rome, where Greg had left off and Catholicism was basically back on track... well except for the massive rift the whole cock-up had created through Europe which almost instantly brought about the Reformation and Protestantism9.