1. Preheat oven to 400 degrees. On a floured surface, roll puff pastry into a 10-by-14-inch rectangle. Trim uneven edges. Place pastry on a baking sheet. With a sharp knife, lightly score pastry 1 inch from edges, creating a border; with a fork, prick dough all over inside border. Bake until golden, 12 to 15 minutes. Let cool completely.

2. Meanwhile, in a medium saucepan, cook pineapple, sugar, and lemon juice over medium-high until pineapple begins to break down, 10 to 12 minutes. Transfer to a blender and puree until smooth.

3. Pour through a fine-mesh sieve back into saucepan, discarding solids, and sprinkle with gelatin. Let stand until gelatin softens, 5 minutes. Bring to a boil over medium-high, stirring, then pour into a medium bowl set in a bowl of ice water. Stir until cool and thickened, 4 minutes. (To store, cover and refrigerate, up to 3 days.)

4. Whisk cream until soft peaks form. Stir pineapple mixture until smooth; fold in whipped cream until combined. Spread on pastry and top with mango and coconut.