Combine cranberries, cherries, sugar, tangerine juice, tangerine zest, star anise, cinnamon stick, and whole cloves in a heavy-bottomed saucepan over medium heat. Bring to a simmer. Reduce heat to low and cook, stirring occasionally, until cranberries are all popped, about 15 minutes.
Remove from heat and let cool to room temperature. Discard star anise, cinnamon stick, and whole cloves, and refrigerate until needed.
