Bring first 6 ingredients to boil in small saucepan over medium heat. Reduce heat to low and simmer until mixture is reduced to 3/4 cup, about 25 minutes. Cool. Strain mixture. Add rum and stir to combine. Refrigerate until cold, at least 1 hour. (Cinnamon-clove syrup can be prepared 2 days ahead. Keep refrigerated.)
Scoop out enough melon balls from honeydew and cantaloupe to measure 4 cups total. Divide melon balls among 6 bowls. Top with syrup and serve.
