1Trim fat from beef. Make cuts about 1/2 inch apart and 1/8 inch deep in diamond pattern on both sides of beef. Place in shallow glass dish. Mix honey, soy sauce, gingerroot and beer in small bowl; pour over beef. Cover and refrigerate, turning occasionally, at least 4 hours but no longer than 24 hours.
2Brush grill rack with canola or soybean oil. Heat coals or gas grill for direct heat. Remove beef from marinade; reserve marinade. Cover and grill beef 6 inches from medium heat about 12 minutes for medium doneness, turning after 6 minutes and brushing frequently with marinade. Discard any remaining marinade.
3Cut beef diagonally into thin slices. Serve beef in pita bread halves with tomato and onion.