Wash the eggs thoroughly and place them together with their shells in a tall container. Squeeze the lemons and limes, pour the juice over the eggs, the eggs must be covered, if necessary use more juice. Put the whole thing in a cool place for 8-10 days, stirring occasionally, the eggshells will dissolve in the meantime.
Put everything through a sieve in a bowl, add sugar and rum, stir until the sugar has dissolved and a cream is formed. Season to taste, use more sugar if necessary. Fill into bottles and let stand for a few more weeks.
I grind the powdered sugar myself with a coffee grinder, the powdered sugar dissolves better!