Bring 1 1⁄2 cups water to a boil. Put tamarind pulp in a small heatproof bowl and cover with the boiling water. Let sit until pulp is soft and mushy, 1 to 3 hours. Using a small potato masher or clean hands, massage and squeeze pulp to break it up. Strain through a fine-mesh strainer set over a medium bowl, pressing with a spoon to extract as much pulp as possible. Chill extract (discard seeds and fibers).
While tamarind is softening, rinse ginger. If it’s young, leave skin on and gently scrub; otherwise, peel. Cut into thin slices, then combine with sugar and 1 cup water in a small saucepan. Bring to a boil over medium-high heat, then immediately remove from burner. Cover and steep 10 minutes. Strain through a fine-mesh strainer and discard ginger (see tip, below). Let cool.
In a small pitcher, mix tamarind extract and ginger simple syrup. Fill 5 tall glasses with ice. Pour about 1⁄2 cup tamarind-ginger syrup into each glass, top with equal amount (or more) of club soda, and stir.
MAKE AHEAD Tamarind extract, up to 4 days, chilled. Tamarind-ginger syrup, up to 1 week, chilled.
SUNSET TEST KITCHEN TIP Leftover ginger, simmered with water, makes an energizing hot drink you can sip like tea; add honey and lime if you prefer.