preheat broiler with rack 3 to 4 inches from element. Line a baking sheet with foil with a rack on top. Lightly coat rack with nonstick spray.
Whisk cornstarch and 2 tablespoons lime juice in a small bowl. Set aside.
Heat honey, orange juice, rum, remaining 2 tablespoons lime juice, garlic, and ginger in a medium pan over high until boiling. Whisk in cornstarch mixture and cook until thick, 1 to 2 minutes. Remove sauce from heat and season with salt and pepper. Set aside to cool.
Stir in cilantro and reserve 1/2 cup sauce for basting and remaining for dipping.
Flip shrimp and baste, with sauce. Broil shrimp until pink, curled, and opaque, one or two minutes more, then serve with dipping sauce. Makes 10 servings