Using a small melon baller, scoop the cantaloupe and the honeydew into balls or use a knife to cut melons into cubes. (You should have about 4 cups cantaloupe and 2 cups honeydew.) Set melon aside.
In a small saucepan combine pineapple juice, sugar, and ginger. Bring to boiling, stirring until sugar dissolves. Reduce heat and simmer, uncovered, over medium heat for 5 to 7 minutes or until the mixture is the consistency of a thin syrup. Remove from heat; cool. Transfer syrup to a storage container. Add 2 cups of the cantaloupe pieces and all of the honeydew pieces. Cover and chill overnight.
Meanwhile, in a blender container or food processor bowl cover and blend or process strawberries until smooth; remove and set aside. Cover and blend or process remaining 2 cups cantaloupe pieces until smooth. In a large mixing bowl stir together yogurt and sour cream. Add pureed strawberries, pureed melon, and milk; stir until combined. Cover and chill overnight.
To serve, drain melon balls, reserving syrup; stir reserved syrup into chilled soup. Ladle soup into bowls; top with melon balls. If desired, garnish each serving with carnation petals. Makes 8 to 10 servings.