Preheat the oven to 350 degrees F.
Place the puff pastry on parchment lined baking sheets, 1/2-inch apart. Place parchment on top of the dough and then on a wire rack or baking sheet on top. Bake until golden brown, about 15 minutes. (If you are using a baking sheet on top, you will have to remove the baking sheet to see if the puff is done.) Cool to room temperature. Slice each piece of puff pastry in 1/2 horizontally along the 4-inch side.
Whisk the cream, sugar, vanilla extract, and rum in a large bowl until soft peaks form.
Slice the bananas 1/2-inch thick on a bias. Place them in a bowl and mix in the lime and orange juices.
Place some bananas and a little rum cream on each of 6 plates. Place a piece of puff pastry on top. Place the remaining bananas and rum cream on the puff. Drizzle warm caramel sauce over and around the bananas. Top with a second piece of puff. Serve immediately.