Rinse and pick over the cranberries, discarding any bruised or mushy fruit. Chop the mandarins coarsely, including the rind, but discarding the seeds. Combine all ingredients in your food processor and process until it forms a slightly coarse relish.
Chill and serve. You can make this relish up to 3 days in advance.
You can culture it! You can culture this relish to make a naturally fermented cranberry relish full of beneficial bacteria. Replace the apple juice with kombucha, and leave out the sugar. Seal the relish in a tight jar (like this one) up to 3 days. Transfer it to a mixing bowl, stir in the sugar, and serve.