Rinse the tuna steaks and pat dry. Arrange them in a shallow dish. In a small bowl, combine the balsamic vinegar, ¼ cup sesame oil, ginger, cilantro, green onions and sugar and mix well. Pour over the tuna, turning to coat. Marinate, covered, in the refrigerator for 1 hour, turning occasionally. Drain, reserving the marinade.
Preheat the grill to high-heat for 10 minutes. Place the tuna steaks on the grill rack. Grill 3 to 4 minutes per side or until the tuna flakes easily.
PREPARE THE MUSHROOMS:
In a skillet, heat the sesame oil and sauté the mushrooms until brown on both sides. Stir in the reserved marinade. Cook until heated through.
SERVE:
Arrange the tuna steaks on a serving platter. Pour the mushroom sauce over the steaks.