I am trying to think of a scenario in 1948 where (1) Truman falls just short of an Electoral College majority (very plausible, requiring just a few votes to be changed in Ohio and California) and therefore the race goes into Congress, the presidency to be determined by the House (on a one-delegation-one-vote basis) and the vice-presidency by the Senate. (2) It is the new, 81st Congress which decides (this was the clear intention of the Twentieth Amendment). (3) As in OTL, the majority of the new House delegations are Democratic, so Truman is elected president, but (4) the Republicans narrowly retain control of the Senate, so Earl Warren is elected vice-president. That's the hard part--the 1948 elections left the Senate with 54 Democrats and only 42 Republicans. Three of the Democratic victories were narrow--in Delaware, Idaho, and Kentucky--but if the Republicans won all three, the Senate would still be 51-45 Democratic. https://en.wikipedia.org/wiki/United_States_Senate_elections,_1948_and_1949 Even if in addition you have Robert F. Wagner resigning for health reasons earlier than in OTL and being at least temporarily replaced by a Dewey appointee, it's still 50-46 D. Maybe have the Democrats nominate a hack instead of Paul Douglas in Illinois? But even if that results in the Republicans holding that seat, it's still 49-47 D. Getting a Republican Senate in the 81st Congress (without a clear Republican victory in the presidential election) requires, if not ASB's, at least too many POD's for my comfort. (Or else you have to have the Republicans win even more Senate races in 1946.)
Anyway, if it somehow happens, the President and Vice-President, though of different parties, could get along quite well. (For one thing, they shared an attachment to Freemasonry: http://www.trumanlibrary.org/photographs/view.php?id=13880) To Truman, Warren was a good man who was in the wrong party for accidental reasons (Warren himself acknowledged that "I was a Republican simply because California [had been] an overwhelmingly Republican state"). Truman said to a California crowd in June 1948, "Your governor pursues forward-looking, liberal policies. He's a man of sense and a man of ability. The facts of the case are he is a real Democrat and doesn't know it." https://books.google.com/books?id=2oS_DhU7zfkC&pg=PA183
And of course there is always the possibility of the Puerto Rican gunmen succeeding in killing Truman, whereupon Warren becomes president...
Anyway, if it somehow happens, the President and Vice-President, though of different parties, could get along quite well. (For one thing, they shared an attachment to Freemasonry: http://www.trumanlibrary.org/photographs/view.php?id=13880) To Truman, Warren was a good man who was in the wrong party for accidental reasons (Warren himself acknowledged that "I was a Republican simply because California [had been] an overwhelmingly Republican state"). Truman said to a California crowd in June 1948, "Your governor pursues forward-looking, liberal policies. He's a man of sense and a man of ability. The facts of the case are he is a real Democrat and doesn't know it." https://books.google.com/books?id=2oS_DhU7zfkC&pg=PA183
And of course there is always the possibility of the Puerto Rican gunmen succeeding in killing Truman, whereupon Warren becomes president...