Wash clementines thoroughly and cut crosswise and into quarters or eighths. Combine clementines with sugar and orange juice in a medium pot over medium high heat and cook, stirring occasionally, until sugar dissolves. Decrease heat to low and cook until clementines are soft and tender, about 30 minutes.
Add cranberries, ginger, cinnamon, star anise, and cardamom and cook, stirring occasionally, another 10 minutes or until cranberries are soft; the sauce will thicken as it cools. Let cool and chill until ready to serve.