Tennessee voter registration forms do not ask the applicant to choose a party, and Tennessee is considered an open primary state. But in the May 3, 2022 primary for local partisan office, a Tennessee voter in Madison County was threatened with prosecution because he is known to be a Democrat, yet he voted in the Republican primary. The voter, Gabe Hart, describes this experience here. As he relates, the Secretary of State, Tre Hargett, in a speech, agreed with Madison County election officials and said insincere voters cannot vote in primaries.
See this commentary from the editor of the Jackson Sun, who disagrees with the Secretary of State.
Tennessee has its primary for federal and state office on August 4, 2022, so this issue may come up again. Thanks to Michael Drucker for this news.