Their title is simply \"Pope\" (not only because they claim to be the true pope; historically, it can be hard to distinguish between legitimate popes and antipopes). An antipope has two of the powers that popes have: excommunication and taxation.
Your pope will accept most papal requests from you, leaving you with only the piety cost and the relations penalty. You can now request claims on counties and duchies, even outside your realm if you successfully press your pope's claim. You can also excommunicate someone you want to imprison, plot against, or steal courtiers from (using \"invite to court\").
Note that while a vassal pope will always agree to your requests, they will still charge double piety and request investiture law change if you're using free investiture, and incur the normal opinion penalties if you overuse his services. Also, at negative opinion, a vassal pope will simply not pay taxes instead of a gradual reduction. (Granted, this is still a tolerable scenario, considering the fact that wealth is not leaving the realm, but remaining in the pope's hands to be used at his discretion.) Remember that even if you do not use a vassal pope's services, you'll be protected from hostile actions (e.g. excommunication) simply by being the pope's liege. With Holy Fury, a vassalized Pope may ask for independence as a request for a coronation. To counter this, set the realm's Investiture Laws to \"Free\" so that the Pope can request for a law change rather than independence.
You can terminate your antipapacy, if your antipope has the status of a count and is a direct vassal of yours, by putting him under the control of one of your dukes; this is easy to do if you make your antipope in a de jure duchy-viceroyalty.
The Papal State is the manifestation of the temporal power of the pope. Beginning with Urbino and Perugia as vassals and with the provinces of Roma, Ancona, Spoleto, Terracina and Avignon, it is relatively large and prosperous compared to other states in the region. Despite this, it is often subject to early invasion by neighbouring states such as Florence and also Provence which often seeks to acquire the landlocked and isolated province of Avignon with which it shares a border. Naples, while not posing any immediate threat due to its starting situation as a lesser partner in a personal union, will often seek to conquer the Papal State should it ever break free from its personal union with Aragon, which usually happens after Aragon's first ruler dies.
Appoint cardinal within the Papal state is expensive but should be done at least once at the start of the campaign before the price gets too high (the price is based on number of cardinals in the world AND in the Papal state). To gain the other six cardinal spend Curia treasury to appoint cardinals in other countries that the Papal state will conquer soon, but not too soon, since that papal state gets a five year peace with the target country. This way the Papal state will have seven cardinals with in the first decades and already have over 100 Invested influence which should be enough to gain Papal control over the second pope.
If it was a regular (not a pope) vassal, I could try to play with inheritance: ensure my heir has a claim on a dutchy and kill the pope and other successors, but... I believe because the duchy government type is a 'Theocracy' I can't inherit it and the pope is going to be elected...
I do think it's worth noting, however, that this situation is not necessarily a bad one. A powerful theocratic vassal can be a nuisance if they like the pope more than you, but has its benefits. With Papal investiture, they receive bonuses to their cardinal eligibility, which will give you influence with the Papacy (especially if they become pope). With free investiture, they can be used for coronations, earning you a decent bonus without having to meet the Pope's demands.
With the exception of Martin I, no pope during this period questioned the authority of the Byzantine monarch to confirm the election of the bishop of Rome before consecration could occur; however, theological conflicts were common between pope and emperor in the areas such as monothelitism and iconoclasm.
Greek-speakers from Greece, Syria, and Sicily replaced members of the powerful Roman nobles in the papal chair during this period. Rome under the Greek popes constituted a \"melting pot\" of Western and Eastern Christian traditions, reflected in art as well as liturgy.
The continuing power of appointment of the Byzantine emperor can be seen in the legend of Pope Gregory I writing to Constantinople, asking them[who] to refuse his election. Pope Boniface III issued a decree denouncing bribery in papal elections and forbidding discussion of candidates for three days after the funeral of the previous pope; thereafter, Boniface III decreed that the clergy and the \"sons of the Church\" (i.e. noble laymen) should meet to elect a successor, each voting according to his conscience. This abated factionalism for the next four successions, each resulting in quick elections and imperial approval.
The prestige of Gregory I ensured a gradual incorporation of Eastern influence, which retained the distinctiveness of the Roman church; Gregory's two successors were chosen from his former apocrisiarii to Constantinople, in an effort to gain the favor of Phocas, whose disputed claim to the throne Gregory had enthusiastically endorsed. Pope Boniface III was very likely of Greek extraction, making him the \"Easterner on the papal throne\" in 607 (many authors incorrectly regard Pope Theodore I, who reigned from 642 to 649, as the first Eastern pope of the Byzantine papacy). Boniface III was able to obtain an imperial proclamation declaring Rome as \"the head of all the churches\" (reaffirming Justinian I's naming the pope \"the first among all the priests\"), a decree Phocas intended as much to humiliate the Patriarch of Constantinople as exalt the pope.
It was regarded as mandatory of a pope-elect to seek the confirmation of his appointment from Constantinople before consecration, often resulting in extremely lengthy delays (Sabinian: 6 months; Boniface III: 1 year; Boniface IV: 10 months; Boniface V: 13 months), due to the difficulty of travel, the Byzantine bureaucracy, and the whims of the emperors. Disputes were often theological; for example, Severinus was not consecrated for 20 months after his election due to his refusal to accept monothelitism, dying only months after he finally received permission to be consecrated in 640. When Greek Pope Theodore attempted to excommunicate two Patriarchs of Constantinople for supporting monothelitism, imperial troops looted the papal treasury in the Lateran Palace, arrested and exiled the papal aristocracy at the imperial court, and desecrated the altar of the papal residence in Constantinople.
Theodore was Greek-Palestinian, the son of the bishop of Jerusalem, chosen for his ability to combat various heresies originating from the East in his native tongue. As a result of Theodore's ability to debate his adversaries in their own language, \"never again would the Papacy suffer the sort of embarrassment that had resulted from Honorius's linguistic carelessness\". Theodore took the nearly unprecedented measure of appointing Stephen of Dor as apostolic vicar to Palestine, with the intent of deposing the Monothelite bishop successors of Sergius of Joppa. Theodore's deposition of Patriarch Pyrrhus ensured that \"Rome and Constantinople were now in schism and at open war\" over the Christology that would characterize the Christian empire. A Greek pope excommunicating the Patriarch no doubt proved a \"distressing spectacle\" for the emperors intent upon restoring religious unity. Theodore's boldness attests to:
According to Eamon Duffy, \"one of the worst elements in Martin's suffering was the knowledge that while he still lived the Roman Church had bowed to imperial commands, and had elected a new pope\", Pope Eugenius I. According to Ekonomou, \"the Romans were as prepared to forget Pope Martin as Constans II was relieved to see him removed to the remote northern shores of the Black Sea\". Thirty years later, the Sixth Ecumenical Council would vindicate the council's condemnation of Monothelitism, but not before the synod \"ushered in the period of Rome's \"Greek intermezzo'\".
The inhabitants of both East and West had \"grown weary of the decades of religious warfare\", and the arrest of Martin I did much to dissipate the \"religious fever of the empire's Italian subjects\". Rapprochement within the empire was viewed as critical to combatting the growing Lombard and Arab threat and thus no pope \"referred again to Martin I\" for seventy-five years. Although the Roman uneasiness of electing a successor while Martin I lived and the Byzantine desire to punish Rome for the council caused the immediate sede vacante to last fourteen months, the next seven popes were more agreeable to Constantinople, and approved without delay, but Pope Benedict II was impelled to wait a year in 684, whereafter the Emperor consented to delegate the approval to the exarch of Ravenna. The exarch, who, invariably, was a Greek from the court of Constantinople, had the power to approve papal consecration from the time of Honorius I.
Justinian II first sent a magistrate to arrest John of Portus and another papal counselor as a warning, and then dispatched his infamous protopatharios Zacharias to arrest the pope himself. Justinian II attempted to apprehend Sergius I as his predecessor had done with Martin I, underestimating the resentment against imperial authority among those in power in Italy, and the Italian-born troops from Ravenna and the Duchy of the Pentapolis mutinied in favor of Sergius I upon their arrival in Rome. Not long after, Justinian II was deposed in a coup (695). However, the thirteen revolts in Italy and Si