Ben,
It is possible they did. That did happen in 1863 at Chickamauga for two consolidated Tennessee regiments. A new 15 star Southern Cross flag of the 7th Texas Infantry that was taken at Franklin by an officer of the 24th Wisconsin (not Arthur McArthur) turned up over the last year and that regiment was also part of Cleburne's Division and would have had a Cleburne/Hardee flag as well but that is within one regiment. Were both carried at Franklin? Not sure as yet.
Greg Biggs