Whisk cornstarch with 2 tbsp. lime juice and set aside.
Heat honey, orange juice, remaining lime juice, rum, butter,garlic, and ginger over high heat until boiling. Add cornstarch mixture and let it thicken. Remove from heat.
Preheat broiler. Line a baking sheet with foil and place a rack on top. Arrange shrimp on rack and brush with sauce. Broil until opaque. Turn shrimp and brush with sauce. Broil until lightly browned. Top with cilantro and serve with remaining sauce as dip.