Crack brittle shells from tamarind pods; peel off shells and fibers. Put fruit and 2 cups water in a medium saucepan, cover, and bring to a boil. Reduce heat and simmer 5 minutes. Remove from heat and let stand, covered, 20 minutes, stirring occasionally. Strain mixture into a bowl, pushing pulp through; discard contents of strainer.
Meanwhile, in another medium saucepan, sauté onion in oil until golden, 6 to 8 minutes. Add tamarind pulp and remaining sauce ingredients. Bring to simmering over medium-high heat; then reduce heat and simmer, partially covered, 15 minutes to blend flavors, stirring occasionally. Purée sauce in a blender until smooth.
* Find at Latino markets and online (sold as "fresh tamarind"), and look for smooth, soft-looking fruit beneath the brittle shells.
Make ahead: Up to 3 weeks, chilled.