Using a sharp knife, peel the oranges, removing all of the bitter white pith. Working over a bowl, cut in between the membranes to release the sections. Squeeze the juice from the membranes into the bowl. Cut the orange sections into 1-inch pieces.
In a large saucepan, combine the cranberries and water with 1/4 cup of orange juice from the bowl and bring to a simmer over moderate heat. Cover and cook over moderately low heat, stirring occasionally, until the cranberries have popped, about 10 minutes. Add the vinegar, both sugars, the ginger and orange pieces and cook over moderate heat, stirring occasionally, until thick, about 5 minutes. Let cool slightly, then season with salt and pepper. Serve warm or at room temperature.