Slice steak into 1/2-inch thick slices across the grain.

Whisk together soy sauce, sugar, cornstarch, and ginger in a bowl until the sugar has dissolved and the mixture is smooth. Place steak slices into the marinade, and stir until well-coated.

Heat 1 tablespoon vegetable oil in a wok or large skillet over medium-high heat, and cook steak strips in batches in hot oil so as not to crowd the pan. Cook and stir until well-browned, about 3 minutes, and remove steak from the wok to a bowl. Repeat with remaining steak, adding more oil as needed, and set the cooked meat aside.

Return cooked steak strips to the hot wok, and stir in onion. Toss steak and onion together until onion begins to soften, about 2 minutes, then stir in green pepper.

Cook and stir the mixture until peppers have turned bright green and started to become tender, about 2 minutes more, then add tomatoes.