Place the cranberries in a strainer and rinse thoroughly. Remove and discard any bruised or mushy cranberries.
Simmer the cranberries:
Combine the granulated sugar, light brown sugar, orange juice, pomegranate juice, and water in a medium pot. Turn the heat to medium and stir regularly until the sugar is dissolved and the liquid is bubbly.
Add the cranberries to the pot and turn the heat to medium-high. Once the cranberries begin to boil, reduce the heat to a simmer. Stir occasionally as the cranberries begin to pop and sputter.
Simmer for 25–30 minutes, gently mashing the cranberries as they cook.
To serve:
Once the cranberry sauce is thickened to your liking, turn off the heat and let stand for 20 minutes. Serve warm or refrigerate for at least 2 hours before serving.