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That's one response to his theological position in the practical order.  On the other hand, one might continue to keep his name in their by virtue of his having legal status as pope.  Of course, in the Canon, it does refer to the included pope as being among the "orthodoxi ... cultores catholicae fidei" (orthodox keepers of the catholic faith) ... which status one would rightly reject for Francis.

So, 2V, does this sound like Father Ringrose considers him a "real" pope?

Sedeprivationism is intellectually absurd. Sedevacantism is doctrinally and morally a dead end. Recognize and resist is the only sound position at this time that is easily defended in spite of the mocking insults delivered by posters on this forum.
 
Caiaphas was a heretic.  He denied the bodily resurrection and rejected Jesus as the Christ. He did not thereby loose his office. Even the apostles after Pentecost did not suggest that he lost his office because of heresy. St. Paul recognized and respected the office when he appeared before the high priest in Jerusalem. “And they that stood by said: Dost thou revile the high priest of God? And Paul said: I knew not, brethren, that he is the high priest. For it is written: Thou shalt not speak evil of the prince of thy people” Acts 23:4-5. “Jesus spoke to the multitudes and to his disciples, saying: The scribes and the Pharisees have sitten on the chair of Moses. All things therefore whatsoever they shall say to you, observe and do: but according to their works do ye not” Matt. 23:1-3. This direction can be accurately described as “recognize and resist.” What God has established, only God can overthrow. Every commentary on the parable of the tares (Matt 13:24) the including Rev. George Haydock Commentary, Rev. Cornelius a Lapide’s Great Commentary, and St. Thomas’ Catena Aurea quoting the Church Fathers without exception say the tares refer to heretics.  Those who demand that heresy precludes anyone from the office want to make themselves the “Lord of the Harvest.” Heresy precludes only be canonical laws, not by the nature of heresy itself.  It is a question of law.  It is ironic indeed that those making themselves the “Lord of the Harvest” end up with the tares.
 
No Catholic is required to do more with a heretical pope than the man born blind, and if he keeps dogma as his proximate rule of faith, a whole host of problems can be avoided. Those who deny dogma as the proximate rule of faith make the person of the pope their rule of faith and what follows is a host of irreconcilable problems.
 
The sedeprivationists offend the first principles of the understanding. The conciliarist popes are either popes or they are not. They cannot be, and not be, at the same time. If they stand in any way in potential to the office, then they are not popes. To divide the office between degrees of material and formal possession is to destroy the papacy. Separation of form and matter always constitutes a substantial change by definition. It is a dogma of faith that the Church founded by Jesus Christ was founded upon Peter. It is further a dogma of faith that the office will have perpetual successors. The faith is the primary sign and cause of unity in the Church. The pope is only accidentally and secondarily the sign and cause of unity and, since he is not the proximate rule of faith, he is just as much subject to the faith as every baptized Catholic. He does not possess the authority to command obedience to anything in violation of the virtue of Religion which is the virtue under Justice that directly governs obedience.  Any act of obedience to any human authority that offends the virtue of Religion is a sin. Just as the man born blind in John 9 professed the true faith to the Pharisees every faithful Catholic is called upon to do the same today.  It did not require him to deny that authority of the Pharisees because of heresy.  When the pope becomes a heretical Judaizer like St. Peter did, when in his “dissimulation… (he) walked not uprightly unto the truth of the gospel,” he must be “withstood to his face” Gal 2:13-14.
 
Sedevacantism is intellectually, morally and doctrinally a dead-end. They have arrived at a Church that is not just defective in an essential attribute but it has no capacity to ever correct the defect therefore it cannot be the Church founded by Jesus Christ. How do faithful Catholics end up in a position that is manifestly erroneous?  Those Catholics that do not accept dogma as the proximate rule of faith necessarily make the pope their rule of faith. They make him the source of revelation as the revealer of mere ecclesiastical faith and they impose an understanding to the attribute of Indefectibility to mean that the pope possess a personal never failing faith and cannot possible teach error or promulgate unjust laws. They cannot recognize a heretical pope without feeling personally contaminated by his sin. But none of this is so. None of this has been dogmatically defined. These are nothing but theological presuppositions; speculative opinions expressed from men who could not imagine the current crisis of the Church. These opinions in our current situation appear daily more and more implausible.
 
Until the Pope uses his office to engage the attribute of Infallibility that Jesus Christ endowed His Church to bind doctrinal error and immorality upon the Catholic faithful, and sedevacantists produce their own papal claimant, there is no argument against the recognize and resist that does not lead to doctrinal and intellectual error. Our obligation is only to remain doctrinally and morally sound in the faith.  No Catholic is obligated to provide an answer every question but he does have an obligation to avoid obvious errors. “Sufficient for the day is the evil thereof” Matt 6:34.

Drew

The third difference is that the case of a heretical pope is different from that of a heretical bishop. A pope is the living rule of faith for the entire Church, and is infallible in his magisterium (whether solemn or ordinary universal), and is infallible in promulgating universal laws, liturgy, and disciplines. None of these things is true of a bishop of a diocese. I remember as a child that people would often say, “You can’t be more Catholic than the Pope.” Very true. He is the living rule of faith, just as a yardstick is the rule of what is one yard. (source: http://www.mostholytrinityseminary.org/SCSF%20February%202018.pdf)