Preheat oven to 375 degrees F (190 degrees C). Lightly grease two 8x4 inch loaf pans.

Beat together 2 tablespoons of softened butter and cream cheese in a large bowl until fluffy. Slowly beat in sugar until well incorporated. Add the egg and mix well.

Sift together the flour, baking powder, baking soda, and salt in a separate bowl.

Mix bananas, milk, 2 tablespoons rum, lime zest, lime juice, and vanilla extract in a different bowl; stir well.

Pour 1/3 of the flour mixture into the butter mixture and mix well. Mix in 1/2 of the mashed bananas. Continue with another 1/3 of the flour, followed by the remaining banana mixture, and finally the remaining flour. Fold in 1/4 cup pecans and 1/4 cup coconut flakes.

Pour the mixture into the loaf pans, and bake in the preheated oven until a toothpick inserted into the center comes out clean, about one hour. Cool for 10 minutes in the pans, then transfer to wire racks to cool completely.

Prepare the topping by stirring together the brown sugar, remaining butter, lime juice, and 2 teaspoons of rum in a small saucepan over medium-high heat. Bring to a simmer stirring constantly; cook until the sugar has dissolved and the mixture is smooth, about one minute.