Believe it or not, there was talk at the 1968 Democratic National Conventiion of a Eugene McCarthy-John Connally ticket. What happened was that Connally was becoming increasingly critical of Humphrey, who he thought was distancing himself too much from LBJ and the war, agreeing to the abolition of the unit rule, etc. Connally even threatened that if Humphrey kept it up, LBJ might re-enter the race. Patrick Lucey saw Connally's dissatisfaction with Humphrey as an opportunity for McCarthy, and sent Richard Goodwin to negotiate with Connally on a McCarthy-Connally ticket According to Dennis Wainstock,
Election Year 1968: The Turning Point, "If Connally became McCarthy's running mate, Lucey reasoned, the South might desert Humphrey for McCarthy. According to Herzog, 'They played cat-and-mouse on the question of a McCarthy-Connally ticket and Connally did not say no.' Later Goodwin, looking 'mysteriously optimistic,' offered to bet any takers that McCarthy would get the nomination. Although McCarthy must have known about the negotiations, he told his staff 'he would rather lose the nomination' than consider Connally as a running mate."
https://books.google.com/books?id=mj70AgAAQBAJ&pg=PA141
For another account, see Dominick Seabrook,
Eugene McCarthy: The Rise and Fall of Postwar American Liberalism, pp. 210-211:
"By the time the convention finally opened on 26 August, McCarthy was 'certain' that he would not be nominated. He had already told Thomas Finney that he was tempted to pull out and support a dark home, perhaps New Jersey governor Richard Hughes or his old friend Edmund Muskie, who would be 'a real President.' There remained, however, the tantalizing possibility of an alliance with the South to stop Humphrey, the old advocate of civil rights, and McCarthy's aides were keen to strike a deal with the southern establishment. To get the nomination, according to Harry Mc-Pherson, McCarthy was prepared to resubmit the names of Abe Fortas and Homer Thornberry for confirmation to the Supreme Court, and was also 'willing to permit President Johnson to name his Vice-Presidential running mate-—within reasonable limits.' Richard Goodwin, who had returned to McCarthy after Kennedy's death, visited Governor John Connally during the convention and offered to support 'a watered-down Vietnam plank and also [to] let the Southerners choose whomever they want for Vice President' if Texas would support McCarthy. If Connally endorsed McCarthy, he would be offered the vice presidency as well as the right to choose some cabinet and judicial nominees. It seems fairly clear that McCarthy himself was unaware of most of the details, and when the plan was presented to him he balked at the idea of Connally as a running mate. The arrangement did make a certain sense. McCarthy had always been close to the Texans, and offered the prospect of a more passive presidency than the hyperactive Humphrey. He had a long congressional history of compromise and accommodation, not to mention close personal links with the South. As Tom Finney reflected, McCarthy's vision was 'more compatible with a lot of the leaders in the south than Hubert Humphrey's view of the presidency.' Had it not been for his campaign in 1968, in fact, McCarthy might well have reamined a southern faviorite..."
https://books.google.com/books?id=wMqSzTPXl7QC&pg=PA210
Anyone see any chance of this actually happening? Given that (1) in 1972, John T. Amos, Humphrey's envoy, give Wallace the impression that Humphrey was favorable to a Humphrey-Wallace ticket (though whether Humphrey really authorized him to promise Wallace the vice-presidency may be doubted) and (2) Ronald Reagan in 1976 offered the vice-presidency to the then generally liberal Richard Schweiker (and John Sears wanted Reagan to offer the vice-presidency to Nelson Rockefeller!) we should not be surprised by seemingly incongrous ideological combinations being at least seriously considered in national conventions in that era. Especially by underdog candidates getting desperate (or at least their aides).
Sure, "I think Humphrey is getting too soft on the peaceniks, so I'll serve as a real peacenik's running mate" would be a strange thing for Connally to (in effect) say--yet according to reports, he didn't reject Goodwin's offer outright...
I just don't think that McCarthy would ever accept the idea, though. He would realize that however close he had once been to LBJ and the Senate Southern Democrats, his appeal in 1968 was based on being anti-LBJ and anti-war, and that his followers would view his acceptance of Connally as a sell-out.
(And yet, a northen liberal Democrat who had opposed the war did eventually offer the vice-presidency to a Vietnam-war-supporting LBJ man from Texas--in 1988... But of course the war was hardly a pressing issue by then. )