Preheat the oven to 350°F (175°C). Grease and flour a 9-inch round cake pan or line with parchment paper.
In a medium bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, nutmeg, and ginger until well combined.
In a large bowl, beat the granulated sugar, brown sugar, and eggs together until smooth and pale, about 2 minutes.
Add the vegetable oil, Greek yogurt, and vanilla extract to the egg mixture and mix until fully incorporated.
Fold the dry ingredients into the wet ingredients gently until just combined; do not overmix.
Fold in the grated carrots, mashed bananas, chopped walnuts, and shredded coconut until evenly distributed.
Pour the batter into the prepared cake pan and smooth the top with a spatula.
Bake for 40 to 45 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and let it cool in the pan for 15 minutes, then transfer to a wire rack to cool completely.
While the cake cools, prepare the frosting: beat the softened cream cheese and butter together until smooth and creamy.
Gradually add the powdered sugar and vanilla extract, beating until the frosting is smooth and fluffy.
Once the cake is completely cool, spread the cream cheese frosting evenly over the top.
Slice and serve. Store any leftovers covered in the refrigerator.