Line a 9-by-5-inch loaf pan with a double layer of foil leaving a 2-inch overhang on all sides.
For the mango nice cream: Put the mango, bananas, honey and vanilla in a food processor and process. Hang on, your food processor may jolt in the very beginning! The bananas will look crumbly at first, then gooey and eventually light and creamy, just like ice cream. Stop the food processor frequently to scrape down the sides of the bowl and break up any clumps. Transfer to the prepared loaf pan and spread evenly with an offset spatula and freeze. Rinse and dry the food processor bowl and blade.
For the strawberry nice cream: Put the strawberries, bananas, honey and vanilla in a food processor and process until smooth and creamy just like the mango nice cream. Spread evenly on top of the mango nice cream with an offset spatula and freeze. Rinse and dry the food processor bowl and blade.
For the pineapple nice cream: Put the pineapple, bananas, honey and vanilla in a food processor and process until smooth and creamy just like the other two nice cream flavors. Spread evenly with an offset spatula on top of the strawberry nice cream and freeze until completely solid, 8 hours and up to overnight.
To serve: Let the nice cream cake sit at room temperature for 5 minutes to thaw slightly. Use the foil overhang to lift and remove the cake from the pan. Place upside down on a serving platter so the mango layer is on top.
Combine the chocolate and coconut oil in a small microwave safe bowl and microwave until melted, about 1 minute. Stir until smooth. Pour over the top of the nice cream cake, spreading and smoothing with an offset spatula to cover the top and so that some of the chocolate drips down the sides. Freeze until the chocolate is hard and set, about 10 minutes. Cut into 1-inch-thick slices.
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