Whisk up anchovies and 3 tablespoons EVOO, poured slowly. Add tomatoes, capers, olives, parsley, red onion, basil, garlic and red pepper flakes; stir and let stand 30 minutes.
Heat a griddle or cast-iron skillet over medium heat. Drizzle fish with remaining olive oil, about 1 tablespoon and season with salt and pepper. Cook fish 2 minutes on each side for rare, up to 4 minutes on each side for more opaque fish. Slice the tuna, arrange on plates and top with lots of raw sauce.