In a shaker, combine 4 cucumber slices, 3 lime wedges, and 2 perilla leaves that have been torn apart. Muddle until crushed and juices/oils are released.
Add 3/4 oz of agave nectar (or your choice of sweetener), 2 oz white rum, and ice. Shake vigorously for 5-10 seconds.
Double strain over fresh ice into a glass, then top with 1 oz of club soda.