Heat oil in large, heavy pot over medium. Add ginger and garlic. Cook, stirring occasionally, until softened, about 5 minutes. Raise heat to medium-high.
Add meat; brown all over. Stir in salt, then tomatoes. Lower heat to medium. Cook, stirring occasionally, 5 minutes.
Add water. Raise heat to high; bring to boil. Lower heat to medium. Simmer, partially covered, 10 minutes. Stir in spinach. Cook 1 minute.
Meanwhile, bring large pot salted water to boil over high heat. Add noodles; cook as per package instructions until tender. Drain. (If they're ready too soon, toss with a little hot water or oil to coat all surfaces and prevent from sticking.).
Divide noodles among 4 large bowls. Divide soup evenly over each. Top each with 1 heaping tablespoon cilantro.
Serve with remaining cilantro, vinegar and soy sauce passed separately.