Preheat the oven to 350 degrees F (175 degrees C). Butter a 9x5-inch loaf pan.
Mix flour, baking soda, and salt together in a bowl.
Beat sugar and butter together in a large bowl with an electric mixer until smooth and well blended. Add eggs, one at a time, beating well after each addition. Add bananas, yogurt, and rum; beat until blended. Add flour mixture and beat on low speed or by hand until just moist. Stir in fresh coconut and pour into the prepared loaf pan.
Bake in the preheated oven until a toothpick inserted in the center comes out clean, about 1 hour. Cool in the pan for 10 minutes.
Meanwhile, whisk together powdered sugar, Key lime juice, and coconut flakes for glaze in a bowl. Drizzle over warm bread.
Remove warm bread from the pan and place on a wire rack. Drizzle glaze over top and allow to cool completely before cutting, about 30 minutes.