The advertisement to the right appeared in many places from the summer of 1990 until 1992.  However, the story does not start with this ad.

On 18 May, 1985 Louis Hubert Remy, Mr. de la Franquerie Francis Dallais met with Giuseppe Cardinal Siri.  According to the report published in various places by Mr. Remy, Siri said: No one has left the conclave.  When questioned whether he had been elected in 1963, Siri merely said: I am bound by the secret.  One presumes the secret of the conclave.  He repeated this twice.  This is the first report to circulate that possibly someone other than Montini was elected in 1963.
Gary Giuffre from near Houston, Texas then launched into writing a book Exile of the Pope Elect, which proposed that Siri was elected in 1963 and in both 1978 elections.  This appeared in part in Fr. Dan Jones Sangre de Cristo Newsnotes.
In the Spring of 1988, Gary Giuffre sent Fr. Peter Tran Van Khoat with $15,000 to Italy to meet with Siri and with Wojtyla.  Fr. Khoat paid $5,000 to get into a position to see Wojtyla as he walked from one place to another.  Fr. Khoat had a copy of a presentation Giuffre had compiled which demonstrated the invalidity of Wojtyla's election, as well as that of Montini and Luciani, to be presented to Wojtyla.  Khoat failed to present the document, although he did get close to Wojtyla.  Khoat did see Siri.  in the fall of 1998 and spring of 1989, he reported that Siri refused to comment directly on whether or not he had been elected Pope.
In April 1989 shortly before Siri's death, part III of Exile of the Pope Elect appeared in Sangre de Cristo Newsnotes.  It is a good summary of the theory as it was being presented at this time.  Basically the theory was that Siri was elected in 1963, then threatened with massive persecution, possibly even a nuclear bomb on Rome if he allowed himself to be presented to the world as Pope.  Siri either resigned or allowed the election to continue and produce an antipope to oppose his papacy.  Some also proposed that before he could accept, the threat was made and he then allowed voting to resume.  These people claim that because his rejection was coerced it was invalid, since it was not a free act.  So he was the pope elect, because he had not been allowed to freely accept or reject the election.
This happened again in both 1978 elections, according to the theory.  However, the only reports circulating in 1978 were that Lefebvre had received three votes, and since these were the only valid votes, he was now pope.  However, Lefebvre never acted on this theory, nor did it live into the '80's. 
On May 2, 1989, Giuseppe Siri died, and the theory began to mutate.  Fr. Khoat now claimed that Siri had been elected Pope in 1958.  This is reported by Jim Condit, which is reproduced on the internet. 
In 1990 part V of The Exile of the Pope Elect appeared.  Along side this Fr. Khoat announced that the Cardinal Camerlengo was calling for a Papal Election. Bishop Donald Sanborn reports that a Vietnamese priest (probably Khoat) claimed that Siri consecrated him bishop and appointed him as Cardinal after completing the celebration of the Novus Ordo Missae.
Mr. Gary Giuffre has recently launched a website in order to present the claim that Giuseppe Siri was elected Pope.
David Hobson has also launched a website, and there are others that are supporting the theory that Siri is Pope.  Simply type in Pope Gregory XVII Siri and search.  (Pope Gregory XVII will get you two other men, as well, the Canadian and the one who recently died in Palmar de Troya, Spain.)
Reports vary as to what happened after Siri's death.  Some claim he appointed a successor, possibly Archbishop Pintenello, who also allegedly consecrated Pope Linus II aka Victor Von Pentz, who was elected in 1994 in Assisi, Italy.  Since Archbishop Pintenello is also now dead, these people claim he also appointed a successor.  No evidence is presented to support this, and Fr. Khoat's declaration that Siri left Cardinals is never addressed.

Are we soon to see Pope Innocent XIV? Time will tell.

In february, 2007, David Hobson reports that Tran Van Khoat went with several witnesses, now all conveniently deceased, to meet with Siri and witness his admission that he was really Pope Gregory XVII

Curriculum Vitae of Giuseppe (Joseph) Siri aka Pope Gregory XVII
Google