In a large saucepan over high heat, bring broth, ginger, garlic, and 2 to 4 chile rings to a boil. Cover and simmer until broth is infused with flavor, 8 to 10 minutes. Remove ginger, garlic, and chile with a slotted spoon if you like.
Add ham and greens; add more broth, if needed, to cover greens. Bring to a boil, covered, and cook until greens are barely tender, 2 to 3 minutes.
Divide the soup between 4 serving bowls and top each with a few more red chile rings if you like.
*Find Chinese mustard greens in specialty produce markets or farmers' markets.
Note: Nutritional analysis is per serving.