Stem mushrooms. Slice the oyster mushrooms in half if large, and slice shiitake and button mushrooms. Put in a pot with broth and bring to a boil, covered. Reduce heat and simmer until mushrooms are almost tender, about 5 minutes.
Add bok choy, 1 tbsp. ginger, and the chiles and simmer 2 minutes. Remove from heat. Stir in 1 tbsp. ginger, the vinegar, Sriracha, and sesame oil. Top with green onions and cilantro.