Centre a rack in the oven and preheat to 350F.
Butter a bundt pan, dust with flour and tap out the excess.
Whisk together flour, baking soda and nutmeg.
Working in a stand mixer with a paddle attachment, beat the butter, coconut oil, sugars, and salt at medium speed for 3 minutes, until well blended. Scrap down the bottom if needed.
Add the eggs, beating for one minute then add the rum and vanilla extract. With the mixer on low, add the mashed bananas.
Add half the flour mixture. When it's almost incorporated add the Greek yogurt then rest of the flour and mix well until blended.
Using a rubber spatula, scrape down the bowl and mix in the chocolate chips and coconut.
Spoon the batter into the bundt pan, smoothing the top.
Bake for 65-75 minutes, until the top is browned and the sides of the cake easily pull away when gently nudged.
Transfer the cake to a rack and let it rest for 5 minutes, then unmould onto the rack. Let cool to room temperature.
For the icing, in a small bowl mix icing sugar and milk until well combined. Drizzle icing over the cake and sprinkle with toasted coconut. Let the cake sit for 20 minutes before slicing to serve.