Preheat oven to 375 degrees F (190 degrees C). Spray a loaf pan with natural cooking spray.
In a large bowl, mash the bananas together. Mix in the eggs, sugar, applesauce, oil, and vanilla extract until well incorporated.
In a medium bowl, sift together the all-purpose flour, whole wheat flour, cinnamon, baking powder, and baking soda. Add in the salt, and mix the dry ingredients with the wet ingredients.
Fold in the carrots and coconut, but don't over-stir. Pour the batter into the prepared loaf pan, and sprinkle poppy seeds on top if you like.
Bake the bread for 55 to 60 minutes, or until a toothpick comes out quite clean. It's fine if a few crumbs are stuck to the toothpick.
Let the bread cool for 15 to 20 minutes before slicing it. I know it is tempting to want to scarf this down immediately, but the bread tends to be very crumbly when I slice into it while it's still hot. Let it cool down first!
Store leftovers in room temperature for a few days. After that, store it in the refrigerator.