In a medium bowl, combine 1 pint blackberries, 2 medium-large peaches cut into 1-inch pieces, 3 tablespoons granulated sugar, and 1 tablespoon rum. Let sit for about 10 minutes, before adding 2 1/2 tablespoons cornstarch and 1 tablespoon all-purpose flour (or more, see baker's notes) and gently tossing.