Cut the ends off the navel and blood oranges, then cut away the peel and white pith. Working over a bowl, cut along both sides of each membrane with a paring knife to remove the segments, allowing them to drop into the bowl. Discard any seeds. Squeeze the juice from the membranes into a small bowl (you should have about 1 cup of juice).
Bring the reserved orange juice, honey and ginger to a simmer in a small saucepan over medium heat. Cook until syrupy, 8 to 10 minutes. Let cool completely; discard the ginger. Toss the syrup with the oranges and apples.