This CATO article by Andy Craig says the First Amendment freedom of association clause protects the ability of two parties to jointly nominate the same candidate if they wish. It says the U.S. Supreme Court was wrong to uphold a ban on fusion in Timmons v Twin City Area New Party.
UPDATE: a slight error in the he article has been corrected. The story originally said Earl Warren was the nominee of three parties in California in 1946. He was the nominee of the Republican Party and the Democratic Party, but not the Progressive Party. There was no Progressive Party on the ballot in California in 1946. The Prohibition Party was the only third party in California that year, but it had its own nominee.