Combine water, soy sauce, rice wine, brown sugar, ginger, white parts of green onion, garlic, cilantro stems, tangerine peel, star anise, and red pepper flakes in a 5- to 6-quart pot. Boil, then reduce heat and simmer, uncovered, 10 minutes.
Add short ribs and gently simmer, covered, turning occasionally, until meat is very tender but not falling apart, 2 1/4 to 2 1/2 hours.
Let meat stand in cooking liquid, uncovered, 1 hour.
Remove meat to a cutting board and discard bones and membranes, then cut meat across the grain into 1/2-inch-thick slices.
Pour beef broth through a cheesecloth-lined sieve into a bowl and discard solids. Skim fat from cooking liquid and transfer liquid to a 3-quart saucepan. Add chicken broth and meat and reheat soup over moderately low heat.
Meanwhile, cook noodles until tender and steam the bok choi.
Ladle broth over noodles and top with meat, bok choi, green onions, and cilantro sprigs.