Sour Mix: In a large bowl, whisk the egg whites until frothy. Stir in the sugar and mix until incorporated. Add the water, lemon juice, lime juice, and orange juice. Mix well. Store in the refrigerator until needed.
In a shaker, add ice and all ingredients except the dark rum. Shake until well chilled and mixed.
Pour into a glass and top with the dark rum. Add fresh orchid for garnish and enjoy.