Ordinarily, the dark horse Senate campaign of an elusive, retired teacher in a tiny Alaskan fishing village would barely register for Senator Dan Sullivan, the Republican incumbent running for re-election with President Trump’s support and a $7 million war chest.

But that dark horse’s name is also Dan Sullivan.

National Republicans, including the sitting senator, were enraged when Mr. Sullivan announced in late May that he would run for the Senate as a member of their party. They accused him of being a Democratic plant to confuse voters and siphon support from their chosen candidate, who faces a tough challenge from former Representative Mary Peltola, a Democrat.

In a state with roughly 600,000 registered voters, even a small number of Republicans accidentally voting for the other Mr. Sullivan could hand a close race to Ms. Peltola.

On Tuesday, despite Republicans’ best legal efforts, both Mr. Sullivans will appear on the primary ballot — alongside Ms. Peltola and about a dozen other less well-known candidates — in a high-profile Senate race that Democrats say they must win to take control of the Senate. Because of Alaska’s primary rules, the top four vote-getters, regardless of party affiliation, will advance to the general election in November.