Bring tamarind and 2 cups water to boil in small saucepan. Reduce heat; simmer until tamarind is soft, stirring often and breaking up with wooden spoon, about 25 minutes. Transfer to strainer set over bowl. Press on tamarind to extract puree; discard solids in strainer. Cool.
Bring 1 cup water, ginger, and sugar to boil in another small saucepan, stirring until sugar dissolves. Reduce heat and simmer 20 minutes. Cool ginger syrup.
Place 6 orange slices in bottom of large pitcher; mash with muddler or handle of wooden spoon. Strain ginger syrup into same pitcher; discard ginger. Add 1 cup tamarind puree, tequila, lime juice, salt, and 2 cups water; stir to blend. Chill until cold.
Fill 6 tall glasses with ice. Divide tamarind drink among glasses. Top each with 1/2 cup chilled club soda; garnish each drink with 1 orange slice.