A clean 1-liter plastic soda bottle with cap; cheesecloth
Stir 1 tsp. ginger, 1 tsp. sugar, and 4 cups water in a 32-oz. canning jar to combine. Cover jar with cheesecloth; secure with a canning jar band (without lid). Let sit 1 day out of direct sunlight at room temperature.
Stir in 1 tsp. ginger and 1 tsp. sugar and let sit 1 day. Repeat twice more. Mixture (ginger bug) should have bubbles throughout and release more when agitated.
After 3 days, purée grapes in a blender until smooth. Strain, pressing on solids (you should have about 4 cups juice).
Combine juice and ½ cup ginger bug in a large nonreactive bowl. Cover with cheesecloth; secure with a large rubber band. Let sit out of direct sunlight at room temperature, skimming white mold from surface, until mixture is slightly foamy and releases bubbles when stirred, 3–4 days. Strain into bottle, cap, and let sit at room temperature until bottle feels pressurized, about 1 day longer. Chill.
DO AHEAD: Soda can be made 2 weeks ahead. Keep chilled.
