Peel shrimp, leaving tails on; devein, if desired. Thread 4 shrimp onto each skewer. Set aside.
Cook brown sugar in a small heavy saucepan over low heat until melted. Add garlic and next 6 ingredients. Cook 5 minutes or until tamarind paste melts. Remove from heat.
Brush shrimp with olive oil. Grill, without grill lid, over medium-high heat (350° to 400°) for 4 to 6 minutes or until shrimp turn pink, turning once, and basting with tamarind glaze. Serve with Smoky Sweet Sauce.
Note: Tamarind is a tree-growing fruit. Its long pods contain a sweet-and-sour pulp from which a paste is made. Look for tamarind paste in the Mexican or Asian section of your grocery store.