Break open the chiles and discard the stems and seeds. In a medium saucepan, cover the chiles with the water and bring to a boil. Cover the saucepan and remove from the heat. Let the chiles stand, stirring a few times, until very soft, about 1 hour. Working in batches, puree the chiles with their soaking liquid in a blender.

In a large pot, toast the cumin seeds over moderately high heat until fragrant, about 1 minute. Transfer the cumin seeds to a spice grinder and let cool completely. Grind the cumin seeds to a powder.

In the same pot, heat the oil. Add the onions and garlic and cook over moderately high heat, stirring occasionally, until softened, about 6 minutes. Add the ground beef, pork, and lamb and cook, breaking up the meat into coarse chunks, until starting to brown, about 10 minutes. Add the ground cumin and cook, stirring, for 1 minute. Add the chile puree and simmer over low heat for 1 hour, stirring occasionally.

Stir the hominy and its liquid into the chili. Gradually stir in the cornmeal. Simmer, stirring, until thickened, 5 minutes. Season the chili with salt and pepper and serve.

The chili can be refrigerated for up to 3 days. Reheat gently.