To prepare relish: In a medium bowl, pour 1 1/4 cups of the orange juice over cranberries and set aside. In a medium saucepan, heat oil over medium heat. Add onion and cook, stirring frequently, until soft and translucent. Add garlic, jalapeño or cayenne, and ginger and sauté until soft (not browned), about 1 minute. Add cranberries with juice, salt, and sugar or syrup. Bring to gentle simmer, cover, and cook for 10 minutes. Remove lid, reduce heat, and simmer until about half the liquid has evaporated, about 15 minutes more. Remove from heat. Add remaining 1/4 cup orange juice and orange peel, mixing thoroughly.