In a small bowl, mix the lime juice and mayonnaise. Gradually whisk in the oil. Mix in the onion and jalapeño to make a dressing and season to taste with salt and pepper.
In a large shallow bowl, combine the shrimp, peaches, and avocado slices. Add the dressing, cilantro, and basil and gently toss to blend. Add the romaine, toss gently, and serve. Alternatively, divide the romaine among 4 plates, spoon the shrimp-peach mixture over the top, and serve.