On April 24, the chair of the South Carolina Republican Party held a news conference to say that the party has demanded that independent gubernatorial candidate Tom Ervin stop referring to himself as an “independent Republican” in his campaign ads. See this story.
South Carolina does not have registration by party, so whether or not any individual is a party member depends entirely on how that individual defines himself or herself. Ervin has already responded to the threat by saying he will not comply with the party’s demand.
There were many independent candidates on the November 2012 ballot who referred to themselves as Republicans or Democrats. That was because, due to a confusing campaign finance law, many legislative candidates from the two major parties were removed from the primary ballots of their own parties and had to run as independent candidates.