Trim excess surface fat from steaks. Rinse steaks and pat dry; set in a single layer on a plate. In a small bowl, mix half the vinegar and half the olive oil. Rub mixture all over steaks to coat. Let stand for at least 15 minutes, or cover and chill up to 4 hours.
Meanwhile, rinse, core, and coarsely chop tomatoes. In a bowl, mix tomatoes, basil, garlic, and remaining balsamic vinegar and olive oil. Add salt to taste.
Lay steaks on an oiled barbecue grill over a solid bed of hot coals or high heat on a gas grill (you can hold your hand at grill level only 2 to 3 seconds); close lid on gas grill. Cook steaks, turning once, until browned on both sides and done to your liking in center of thickest part (cut to test), 10 to 13 minutes for medium-rare.
Transfer steaks to plates. Let rest in a warm place for 3 to 5 minutes. Add salt and pepper to taste and garnish with tomato-basil relish.