Whisk together the soy sauce, garlic, Mirin, sesame oil, lime juice, and ground ginger in a medium bowl. Set aside.
Slice the steak against the grain and add it to a large bowl. Mix the steak with 1 tablespoon of cornstarch. Set aside.
Heat a large skillet over medium-high heat. Add 1 tablespoon of vegetable oil to the hot skillet.
Add the steak, and sear on all sides for 3 to 5 minutes. Remove the steak to a plate.
Heat the additional tablespoon of vegetable oil in the same skillet. Add the bell peppers, onions, and snow peas. Saute the vegetables until tender, about 5 minutes.
Add the steak and soy sauce mixture to the skillet. Stir and cook until the sauce has thickened, 1 to 2 minutes.
Turn off the heat and stir in the basil.
Serve the stir-fry over rice garnished with cilantro, green onions, and lime wedges.