Place the new potatoes in a steamer over boiling water and cook for 15 minutes or until tender.
Meanwhile, heat a griddle pan. Pat the tuna fillets dry with kitchen paper and cook in the pan for 3 minutes on each side for rare, 5 minutes for medium or 8 minutes for well done.
Remove the potatoes from the steamer. Slice them in half and place in a bowl. Add the spinach, olive oil and balsamic vinegar. Toss and season to taste. Divide the salad between 4 plates and serve with a slice of tuna arranged on the top of each, and a griddled lime wedge for squeezing.