Heat the olive oil in a large Dutch oven over medium-high heat. Add the onions and peppers, cooking until softened, about 8 minutes.
Stir in the garlic, cook for about 30 seconds. Pour in the beer, scraping to deglaze the pan. Add about half the broth (it does not need to be exact), tomatoes, lime juice and about half the hominy. Simmer until the tomatoes have softened.
Transfer to a blender, blend until smooth, return to pot.
Add the remaining broth, remaining hominy, the spices and the raw chicken.
Simmer until the chicken has cooked through.
Remove chicken from the pot, shred using two forks, return to pot. Adjust spices to taste.
Serve warm, allowing guest to garnish as they choose.
Notes
*if you don't have green tomatoes:
remove the green tomatoes and the lime juice from the recipe, replace with 1 lbs tomatillos, husked and quartered.