Bring water and sugar to a simmer, stirring until sugar is dissolved.
Transfer 1/2 cup hot syrup to a bowl and stir in tarragon, then cool. Chill remaining syrup.
Pour tarragon syrup through a very fine sieve into a metal bowl, discarding solids. Purée honeydew in a blender and stir into strained tarragon syrup. Freeze in ice cream maker, then transfer to an airtight container and put in freezer to harden.
Dice enough of plums to measure 1/2 cup and reserve. Coarsely chop remaining plums and purée in 2 batches with 1 cup chilled syrup (total) in a blender.
Pour purée through very fine sieve into a bowl, pressing on and discarding solids.
Stir in diced plums, lime juice, and chilled syrup to taste.
Serve soup immediately, topped with scoops of honeydew ice.