Preheat your oven to 400°F.
Combine the dry ingredients in a medium-sized mixing bowl. In a larger bowl, mash the bananas and beat in the egg whites until the mixture is light and frothy. Blend in the honey (or other sweetener), vegetable oil, orange juice and rum.
Sprinkle in the dry ingredients and blend until just combined. Lightly grease muffin cups and fill two-thirds full. Bake for 18 to 20 minutes.
These muffins won't rise up in the middle because there is such a small percentage of flour in relation to the oatmeal and bran. But because of the banana, they are delicious, moist and chewy, a great way to start a healthy day!