Preheat the oven to 350°F (175°C). Thoroughly grease a 10 x 5 x 3 1/2-inch loaf pan, then set it aside.
In a large mixing bowl, whisk together the flour, granulated sugar, brown sugar, baking powder, baking soda, sea salt, and freshly grated nutmeg until evenly blended.
In a separate large bowl, whisk the eggs with the vanilla extract. Add the mashed bananas and continue whisking until the mixture is smooth and fully combined.
Pour the wet mixture into the bowl of dry ingredients. Stir gently just until combined, taking care not to overmix. Add the melted butter and stir until incorporated. Fold in the drained crushed pineapple and shredded coconut until evenly distributed throughout the batter.
Transfer the batter to the prepared loaf pan and spread the surface evenly. Sprinkle the top with coarse sugar, if using.
Bake until the loaf is golden and a cake tester inserted into the center comes out clean, about 1 hour and 10 minutes.
Let the bread cool in the pan for 20 minutes before removing and slicing.