Preheat the oven to 350 degrees. With a fine sieve, sift together flour and 3/4 cups sugar four times.
In the bowl of a stand mixer on medium speed, beat together egg whites and water until foamy. Add salt, cream of tartar, and vanilla; beat until soft peaks form. Increase speed to medium-high and sprinkle in remaining sugar, 1 tablespoon at a time. Beat until stiff but not dry.
Transfer to a large bowl. In six additions, sift dry ingredients over meringue, folding in quickly but gently.
Pour batter into an ungreased 10-inch tube pan with removable bottom. Smooth top with an offset spatula. Run a knife through batter to release air bubbles. Bake for 35 to 40 minutes, until golden brown and springy to touch.
Invert pan on its legs or over the neck of a glass bottle, and let cool completely, about 1 hour.
Make the frosting: In the heatproof bowl of an electric mixer set over a saucepan of simmering water, combine egg whites, sugar, water, and cream of tartar. Cook over medium heat, whisking frequently, until sugar is dissolved and mixture is foamy, about 5 minutes.
Attach the bowl to a stand mixer fitted with the whisk attachment. Beat the mixture on high speed until glossy and voluminous, about 7 minutes. Beat in the vanilla.
Carefully run a long offset spatula or knife around the inner and outer perimeter of the cake pan to release cake.
Place on a plate, bottom side up. Using a long serrated knife, carefully slice off 1 1/2 inches from the top of the cake, being sure not to break the layer. Set aside. Cut out a 1-inch-wide, 1-inch-deep channel halfway between the center and edge of cake. Spread icing into channel and over entire layer. Sprinkle with coconut.
Gently place reserved layer on top and ice with remaining frosting. Sprinkle with coconut and serve immediately.
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