Stir together 1/3 cup cream, sugar, lime leaves, zest, and a pinch of salt in a small heavy saucepan and bring to a bare simmer over moderately low heat, stirring until sugar is dissolved.
Remove from heat and let steep, covered, 25 minutes.
Sprinkle gelatin over water and lime juice in a bowl and let soften 1 minute. Return cream mixture to a simmer, then pour through a very fine sieve into gelatin, pressing on and discarding solids. Stir until gelatin is dissolved, about 1 minute. Gradually add yogurt, whisking, and chill, covered, 30 minutes. (
Mixture will not be gelled.)
Beat remaining 1/3 cup cream in a chilled bowl with whisk until it just holds soft peaks. Fold into yogurt mixture and chill, covered, at least 8 hours. (Mousse will be softly set.)
Blend honeydew, Midori, lime juice, and a pinch of salt in a blender at high speed until smooth, about 1 minute.
Pour through a fine sieve lined with dampened cheesecloth into a bowl.
Let drain, undisturbed, 30 minutes, then discard foam remaining in cheesecloth.
Sprinkle gelatin over water in a small saucepan and let soften 1 minute.
Add 2 tablespoons melon juice and sugar, then cook over low heat, stirring, until gelatin and sugar are dissolved, 1 to 2 minutes. Gradually whisk in 1 1/8 cups melon juice (reserve remainder for another use if desired), then transfer to a bowl and chill, covered, at least 8 hours. (Honeydew water will be barely set.)
Spoon honeydew water onto 4 dessert plates and top with shiso leaves. Spoon mousse onto leaves.
