Preheat your oven to 350°F (175°C). Grease and flour a 13x9 inch baking sheet pan, or line it with parchment paper for easier removal.
In a medium bowl, whisk together the flour, baking powder, baking soda, salt, and shredded coconut. Set this mixture aside.
In a large mixing bowl, beat the softened butter and granulated sugar together until light and fluffy, about 3-4 minutes.
Add the eggs, one at a time, to the creamed butter and sugar, beating well after each addition. Stir in the vanilla extract and lime zest.
Mix the buttermilk and fresh lime juice in a separate bowl. Gradually add the dry ingredients to the butter mixture, alternating with the buttermilk mixture, starting and ending with the dry. Mix until just combined.
Pour the batter into the prepared sheet pan, smoothing the top with a spatula. Bake for 25-30 minutes, or until a toothpick inserted in the center comes out clean.
Once baked, remove the cake from the oven and allow it to cool in the pan for about 10 minutes. Then transfer it to a wire rack to cool completely.
In a small bowl, mix the powdered sugar, lime juice, and lime zest until smooth. Adjust the consistency with more lime juice or powdered sugar if necessary; it should be pourable but not too runny.
Once the cake is completely cooled, drizzle the lime glaze over the top, allowing it to drip down the sides.
Sprinkle the additional shredded coconut on top of the glazed cake for a tropical touch.
Notes
For a decorative touch, garnish each slice with a lime wedge or extra shredded coconut.