Slice the 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 the steak slices into the marinade, and stir until well-coated.
Heat 1 tablespoon of vegetable oil in a wok or large skillet over medium-high heat, and place 1/3 of the steak strips into the hot oil. Cook and stir until the beef is well-browned, about 3 minutes, and remove the beef from the wok to a bowl. Repeat twice more, with the remaining beef, and set the cooked beef aside.
Return all the cooked beef to the hot wok, and stir in the onion. Toss the beef and onion together until the onion begins to soften, about 2 minutes, then stir in the green pepper. Cook and stir the mixture until the pepper has turned bright green and started to become tender, about 2 minutes, then add the tomatoes, stir everything together, and serve.
