In it, but not of it. TPM DC
Lieberman would face serious problems if he were to run again -- either in the Democratic primary to rejoin his former party, or the Republican primary, or as an independent.
After he lost the 2006 Democratic primary, he was then re-elected as an independent with unofficial Republican support. Although he remained in the Democratic caucus -- as he'd pledged to do during that campaign -- he nevertheless campaigned aggressively for John McCain for president in 2008, and spoke at the Republican National Convention. After McCain lost the election to Barack Obama, Lieberman settled back into place in the Democratic caucus.
During the health care reform debate, Lieberman further inflamed liberals by helping to stop the public option proposal, promising to support a Republican filibuster. He did in fact ultimately vote for final passage of the more conservative reform law that did not include a public option -- and which is a bane to Republican voters.
This past October, a survey from Public Policy Polling (D) found Lieberman running in third place in various trial heats with hypothetical Democratic and Republican candidates, with only marginally positive ratings from Republican voters and deeply negative ratings with Democrats and independents.