Preheat the oven to 350 degrees F.
Toss the bread cubes with the melted butter in a large bowl, spread on a large baking sheet, and bake until lightly browned, about 10 minutes. Transfer to a 7 by 11-inch baking dish and set aside.
Beat the eggs in a large bowl. Add the sugar, rum, vanilla, nutmeg, and salt and whisk to combine. Slowly add the milk, whisking constantly. Add the syrup from both canned, drained fruits and whisk to combine.
Arrange the fruit over the toasted bread cubes. Pour the custard evenly over the mixture, dot with the sliced butter, and bake until the custard sets, about 35 minutes. Let rest for 15 minutes.
Spoon the pudding into dessert bowls, ladle 3 to 4 tablespoons of the rum sauce over each serving, and serve.
Whisk together the egg yolks, cornstarch, sugar, and vanilla in a medium saucepan. Slowly whisk in the warm milk and cook, stirring constantly, over medium-low heat until the mixture thickens, 3 to 4 minutes. Strain through a fine mesh strainer into a medium bowl and stir in the rum. Serve room temperature or chilled.