Heat oven to 180°Celsius. Grease a 40cm Bundt tin.
Whisk eggs, lime zest & juice, milk and mashed bananas together until combined in a large bowl. Set aside.
Cream the butter and sugar in a stand mixer until light and fluffy. Add banana mixture and beat on low until well combined. Sift in the flour and baking powder, then stir through shredded coconut. Fold in the wet mixture until just combined. Transfer to the greased bundt pan and bake in oven for 40-45 minutes until cooked through. Its cooked through when a thin skewer inserted in the cake comes out clean. Allow to cool.
Make the icing. Combine the lime rind and half of the icing sugar in a small bowl and mix well. Stir in the lime juice slowly until icing is thick and creamy. Ice the cake and enjoy.