Basil-Matcha Syrup: Combine the matcha powder, basil, sugar, and 3⁄4 cup hot water in a blender and blend on high until smooth and bright green and the sugar is dissolved, about 2 minutes. Strain through a cheesecloth-lined strainer, discard the solids, and let the syrup cool to room temperature. The syrup will keep for 1 week in the refrigerator.
Combine the syrup, lime juice, and coconut milk in a cocktail shaker. Fill with ice, seal the shaker, and shake just to combine, about 3 seconds. Double-strain into a tumbler filled with ice and top with soda water. (This will produce foam, so pour slowly and carefully.) To garnish, sift matcha powder on top of the foam.
Batch for 6: Combine 3⁄4 cup syrup, 1⁄2 cup plus 2 tsp. lime juice, 3⁄4 cup coconut milk, and 2 cups ice in a blender and pulse once or twice, just to combine. Divide among 6 tumblers and top each with soda water. Garnish each with a dusting of matcha powder.