Grease a 10 to 12-inch bundt cake pan (26cm) with 2 tablespoons shortening, making sure every nook, cranny and angle is covered. Dust with flour, shaking off excess. Reserve.
Preheat oven to 350ºF (180°C).
Sift 2 ½ cups all-purpose flour, 2 ½ teaspoons baking powder and ½ teaspoon salt. Set aside
In a large bowl, beat ¾ cup unsalted butter until creamy. Slowly add 1 ½ cups sugar and beat 2 minutes.
Add 3 eggs one at a time, beating well after each addition, and then beat for 2 minutes.
In a small bowl, mix 1 cup whole milk, ¼ cup coconut milk, 2 tablespoons dark rum and ½ teaspoon vanilla.
Beginning and ending with dry ingredients, add them in three parts alternating with liquids in two parts.
Add ⅔ cup shredded unsweetened coconut and stir to combine without beating.
Pour ⅔ of batter into prepared pan, spreading evenly. Scatter mango pieces on top. Cover with remaining ⅓ of batter.
Bake for 40-45 minutes, or until a toothpick or cake tester comes out clean.
Let cool on a wire rack for 15-20 minutes, then carefully invert onto the wire rack. Cool completely.