In a med bowl, mix 1/4c olive oil, lime juice, vinegar, garlic, ginger, and 1/4c cilantro. Add tuna and turn to coat evenly. Let marinate at least 1/2, but the longer, the better.
In a small bowl, mix honey, 2 tbsp olive oil, and 2 tbsp cilantro. Set aside.
Remove tuna from marinade, but reserve marinade. Over med to low grill temp, grill tuna for 6-7 minutes per side, basting frequently with marinade. When fish is almost done, brush both sides with honey glaze, remove from grill, and serve.