Peel half the mango and cut 1/2 -inch-thick lengthwise slices down to the pit. Cut crosswise to release the slices. Repeat the process with the other half, then cut the mango slices into 1/2 -inch dice.
In a small nonreactive saucepan, combine the mango, blueberries, sugar, lemon juice, candied ginger, and lemon zest with 1/4 cup of water. Bring to a boil over high heat. Lower the heat and simmer until the fruit is softened and the liquid thickened, about 15 minutes. Let cool slightly.
Make Ahead: The compote can be prepared up to 3 days ahead; cover and refrigerate. Rewarm before serving.