Preheat the oven to 375°. Line a baking sheet with parchment paper. Unfold the puff pastry on the baking sheet and cut it into two 5-by-6-inch rectangles. Cut each rectangle on the diagonal to form 2 triangles. Refrigerate until the pastry is chilled, about 10 minutes.
Separate the triangles so they're evenly spaced. Bake the pastry for 25 minutes, or until nicely browned and crisp.
Meanwhile, in a saucepan, bring the sugar and water to a simmer over moderate heat. Add the blackberries and cook, stirring, for 1 minute. Add the raspberries and cook for 10 seconds. Stir a few times; transfer to a bowl.
Place the pastry on plates, top with the berries and serve.