In a medium bowl, mix together 1/4 cup olive oil, lime juice, balsamic vinegar, garlic, ginger, and 1/4 cup cilantro. Add tuna fillets, and turn to coat evenly. Let marinate for several hours in the refrigerator.
Preheat an outdoor grill for high heat and lightly oil grate. In a small bowl, mix together honey, 2 tablespoons olive oil, and 2 tablespoons cilantro; set aside.
When grill is hot, reduce heat to low, and place tuna fillets on grate. Close lid, and cook for 1 to 2 minutes. Flip fillets over carefully, and close lid again for another minute to sear fish. Open lid, and continue cooking until barely done, basting frequently with marinade. When fish is almost cooked through, brush the honey glaze over both sides of fish, and remove from grill.