Wrap kumquat seeds, cloves, and cardamom pods, if using, in a piece of cheesecloth; tie with kitchen twine to enclose. Place cheesecloth package in a large saucepan along with kumquats, orange juice, sugar, ginger, star anise, cinnamon stick, Aleppo pepper, and salt. Bring to a boil over high heat; immediately reduce heat to a simmer and cook until kumquats are soft and translucent and mixture is thickened and jamlike, about 1 1/2 hours.
Add cranberries to saucepan and cook until they just begin to pop, about 10 minutes. Remove from heat and let cool completely. Sauce may be kept in an airtight container, refrigerated, for up to 3 days.