The House of Cardinal Rampolla del Tindaro, one of the largest factions in the Catholic Hierarchy. Al who have been members since the 19th century are listed in this one article, with links to the ecclesiastical filed on each at catholic-hierarchy.org
I (anonymity) once entered into e-mail discussion with a couple of people who believed the Cardinal Siri thesis. There are very few such people (almost none), and in fact both of these people were personal friends of the originator of the theory, Gary Giuffre. I concluded that Mr Giuffre was a confidence trickster , and that his theory was a mischievous prank which he played on gullible people. The central claim, that he was pope all the time (and not John XXIII, nor Paul VI, nor John Paul I, nor St John Paul II), is obviously false. One the one occasion that Cardinal Siri was asked about it, he was astonished and denied it. Everything else is sheer invention. The details about white smoke mean literally nothing. The idea that some French cardinals could veto the election of a Pope is nonsense. When I was writing to these people they were not (yet) claiming that this priest Carlo Taramasso was a Cardinal, and I think that claim may have been invented after the time I was writing to them (which was in 2004). In conclusion, the theory is rubbish. In summary : a) The 1958 conclave produced Pope John XXIII, and nobody disputed this b) Since then there have been five more conclaves, and five more Popes c) Siri himself took part in three of them (1963 and two in 1978) d) Neither he, nor any other of the cardinals who voted for him ever claimed Siri was elected Pope e) There is universal agreement as to who the Pope is.