Preheat oven to 350 degrees F (175 degrees C). Line muffin cups with paper liners.
Blend mango, bananas, brown sugar, and juice of 1/2 lime in a blender until smooth.
Whisk flour, baking soda, baking powder, cinnamon, nutmeg, and allspice together in a bowl. Mix butter, eggs, and mango-banana mixture into flour mixture until just combined. Pour batter into the prepared muffin cups.
Bake in the preheated oven until a toothpick inserted in the center of a muffin comes out clean, 25 to 30 minutes. Cool in the pans for 10 minutes before removing to cool completely on a wire rack.
Whisk confectioners' sugar, water, and 1 teaspoon lime juice together until glaze is smooth and thick; spoon onto cooled muffins.