In a bowl, mix the ginger, green onions, jalapeƱo, cilantro, tamari, and sesame oil. Add the tuna and toss gently. Cover the bowl and refrigerate for 1-2 hours before serving.
Cut the avocado and mix with the tuna right before serving. This will prevent the avocado from browning.
Separate the lettuce leaves and arrange on a serving platter. Spread out the tuna mixture and squeeze with lime over the top.