Remove the outer pod from the tamarinds, discard, and place the pulp in a bowl. Cover them with 4 cups of boiling water and let them sit anywhere from 2 to 24 hours.
With your hands, clean the tamarinds of the large seeds and strains/threads. Strain into a colander, placed over a bowl to capture the liquid, pressing with your hands or a spoon to get as much pulp as possible. Place the resulting tamarind thin puree/juice in a saucepan. Bring to a boil, add the sugar, and let it simmer over medium-low heat for 25 minutes or until the juice thickens considerably to a thick syrup consistency.
Squeeze in the zest and fresh juice of the lime, let it simmer for a couple more minutes. Remove from heat and let cool. Refrigerate and store in a tightly lidded container. It will keep for months.