Start by peeling the tamarind pods to remove the outer shell. If the tamarind has visible fibers, gently pull them off as well.
In a large pot, combine the peeled tamarind, 3 cups of water, and the sugar. Bring the mixture to a boil over medium-high heat. Once it starts boiling, reduce the heat and let it simmer, covered, for 8–10 minutes.
Remove the pot from the heat and let it cool for about 10 minutes. Once it's cool enough to handle, use a potato masher (or the back of a spoon) to crush the softened tamarind, helping release all the pulp from the seeds.
Place a fine-mesh sieve over a large bowl or pitcher. Working in batches, press the mashed tamarind mixture through the sieve to extract as much juice as possible. Discard the solids once you've squeezed out all the liquid.
Pour the extracted tamarind juice into a serving jug and stir in the remaining 2 cups of water. Add ice cubes to the pitcher or individual glasses and serve the juice chilled.