run as (something)
To be a candidate in an election for a particular political party. After failing to get the endorsement of the Republican Party, she decided to run as a Libertarian. He was a Democrat for most of his career, but he's running as a Republican on this year's ticket.
run as something
to run for office in a certain party. Do you suppose I can run as an independent? Fred ran as a Democrat and won a seat in the legislature.