Grease and lightly flour a 9 x 13 baking dish. Sift flour, baking powder, and salt together.
Separate the egg yolks and whites. Beat the whites until very foamy and frothy. Set aside. In a large mixing bowl, cream the sugar, egg yolks, and vanilla extract with an electric mixer.
Continue mixing and slowly add the milk. Then start adding the flour mixture, a little at a time, until all the flour is incorporated and the batter is smooth. Finally, use a spatula to gently fold in the beaten egg whites until completely mixed.
Pour batter into a greased and lightly floured 9 x 13 baking dish. Bake for 35 to 45 minutes, or until a toothpick stuck in the middle comes out clean. Allow cake to cool on a wire rack for 20 minutes.
Invert the cake onto a jelly roll pan or a baking tray with a raised edge on it – something to catch the syrup! Use a fork to repeatedly pierce the top of the cake.
Whisk together the sweetened condensed milk, evaporated milk, heavy cream and rum in a mixing bowl, until well blended. Pour syrup over the cake a little at a time, until it soaks through. This is a cake that demands patience! Pour the milk syrup slowly and let it soak in. Use your fork to help it along a bit, piercing here and there through the syrup until it soaks in. Keep doing this until most of the milk syrup is absorbed. Don't worry if there is still a pool of syrup around the bottom of the cake, it will gradually soak in.
Cover the now syrup-drenched cake in plastic wrap and refrigerate for at least three hours before frosting.
Beat the egg whites with the cream of tartar in a large bowl until they form stiff peaks. Set aside. In a separate pan, mix the water, sugar, and corn syrup. Cook over a high heat, stirring constantly until a candy thermometer reads 230°F. Remove from heat. While beating the egg whites with an electric mixer, immediately pour the hot syrup into the egg whites; continue beating for about five minutes. Let frosting cool in refrigerator. With a wet spatula, spread a thick layer of the frosting on the cake. To serve, cut individual squares, top with a red maraschino cherry, and serve cold.