Combine broth and next 7 ingredients (through salt) in a medium bowl, and stir with a whisk. Set aside.
Heat oil in a wok or large nonstick skillet over medium-high heat. Add ginger and garlic; stir-fry 30 seconds. Add bok choy, mushrooms, onions, and shrimp. Stir-fry 3 minutes or until bok choy begins to wilt. Add broth mixture to pan; bring to a boil. Cover, reduce heat, and simmer 10 minutes or until vegetables are tender and shrimp are done. Serve over rice.