Put sugar with 1/4 cup water in a small, deep, heavy saucepan and, without stirring, bring to a boil over medium-high heat. Using a pastry brush dipped in water, brush down sides of pan to dissolve any sugar granules. When sugar has dissolved and begins to darken, 6 to 8 minutes, tilt and swirl it to cook evenly. Cook, swirling pan occasionally, until sugar is a deep reddish brown, about 2 minutes more.
Remove from heat and whisk in about 1 tbsp. of cream (sauce will foam up). Slowly whisk in the rest.
Whisk in rum, let cool to room temperature, and chill at least 2 hours to thicken.
Meanwhile, on a floured work surface and with a floured rolling pin, roll pastry sheet out to a 10- by 14-in. rectangle.
Cut pastry in half to form two 10- by 7-in. rectangles and place halves on a large, rimless baking sheet. Lightly prick pastry halves with a fork, leaving a 1/2-in.-wide border.
Peel bananas, cut into 1/4-in.-thick slices, and arrange on pastry.
Brush borders with egg wash and sprinkle tarts all over with brown sugar.
Bake tarts on bottom rack of oven 15 to 17 minutes, or until crusts are medium golden brown (lift edges of tarts to check doneness of undersides).
Let cool slightly and drizzle generously with caramel sauce.
Serve tarts warm or cool, each cut into 4 to 6 pieces.
