Sede vacantes hold that a cardinal named Siri was elected pope in the conclaves of 1958 and 1963, but was prevented from assuming power by some sort of conspiracy. Of course, as conclaves are secret, only the cardinal electors can state with any authority what occurs in them, and they are sworn not to. Furthermore, Cardinal Siri himself remained obedient to the popes of the Church until his death in 1989, so it seems hardly likely that he considered himself a pope-in-exile. The sede vacantes, like all conspiracy theorists, have their photos (in this case, inlvolving the colors of the different emissions of smoke from the conclave) and bad guys, but they are long on innuendo and short on fact or reason.
Still more illogically, sede vacantes hold that Vatican II and the popes of that time and subsequent to it have somehow been “condemned” by popes who were dead before the council even met! Essentially, the sede vacantes will cherry-pick the statements of popes whom they like and apply the words of these popes to popes who don’t suit their tastes. It’s quite a trick to assert that a dead pope condemns a successive pope from the grave! There are other serious problems with this argument. For one, when He established the papacy upon Peter, Christ assured the invincibility of the Church (Matt 16:18) and if She truly has no pope, She has been conquered and He lied, which He cannot do. Also, the Church teaches that each pope is a successor of St. Peter and holds all of the powers of that office, so the powers of the office do not “dilute” over time (that is, it is erroneous to say that St. Peter was the pope with the most authority, that each of his successors have had less authority than those before them, and that Pope Benedict XVI has still less authority than Pope John Paul II).
What the sede vacante position boils down to is a profound disagreement with the doctrines and implementation of the Second Vatican Council — “modernism,” in their catch-all term. Hence, the mental contortions to discredit the Council and recent popes. There are many legitimate complaints about the implementation of the Council, though the documents themselves are quite profound. Our Holy Father has frequently critiqued the changes-for-the-worse in liturgy and one need only look at the sad decline in practice and numbers of many religious orders to see big problems, but the spiritual-Luddite mentality of the sede vacantes and their ugly rhetoric that often smacks of anti-Semitism (for example) is not the solution!
My admittedly limited dealings with members of this group have been frustrating and, for the reasons I’ve discussed above, I doubt most Catholics will get very far in discussions with the sede vacantes. However, we always have recourse to prayer and that, frankly, is what I recommend for these souls lost in their self-made sea of errors. And, it is also important that we pray that they not lead more troubled Catholics out into the murky waters with them!