Peel and devein shrimp, leaving tails intact; set aside.
Combine ketchup and next 4 ingredients in a small bowl. Microwave at HIGH 5 minutes, stirring after every minute. Let cool. Combine mayonnaise and half of ketchup mixture, stirring well; cover and chill mayonnaise mixture at least 15 minutes. Set aside remaining half of ketchup mixture.
Thread shrimp onto 3 (10-inch) metal skewers. Coat grill rack with cooking spray; place on grill over medium-hot coals (350° to 400°). Place skewers on rack; grill, covered, 3 to 4 minutes on each side or until shrimp turn pink, basting frequently with reserved ketchup mixture. Remove shrimp from skewers. Place on a serving platter, and serve with reserved mayonnaise mixture.