Slice off tops of tomatoes (about 1/2 inch). Scoop out the seeds and ribs, and coarsely chop. Stir together chopped tomatoes, peaches, cucumbers, corn, oil, 1 tablespoon lime juice, salt, and pepper. Let stand for 15 minutes. Taste and adjust for acidity, adding up to 1 tablespoon more lime juice. Stir in basil.
Season inside of tomato shells with salt and pepper. Fill tomatoes with tomato mixture, and garnish with basil sprigs. Serve immediately.