To make the marinade, mix together the rum, oil, garlic, allspice, cayenne pepper and some black pepper. Put the beef in a shallow dish and brush the marinade over both sides of each steak. Cover loosely and leave to marinate in a cold place or the refrigerator for at least 4 hours, preferably overnight.
Before cooking, let the steaks stand at room temperature for 30 minutes. Meanwhile, arrange the sliced fruit on individual plates. Whisk together the oil, lime juice, mint and salt, and set aside.
Place the steaks in a heavy, non-stick frying pan over high heat. Cook for 4 minutes on each side for rare steaks, longer if you prefer them well done. Arrange the steaks on the plates with the fruit. Dribble the dressing over the fruit or serve it separately. Serve at once.
