There are three main versions of this hominy stew, each representing a color of the Mexican flag. Red posole, like this version, is made from dried chiles; our stew features ancho chiles. The green version is made from fresh chiles, and white posole has no chiles.
Heat a cast-iron skillet over high heat. Place chiles in pan; flatten with a spatula. Cook 10 seconds on each side or until blackened. Combine toasted chiles and 2 cups boiling water in a bowl; let stand 10 minutes or until soft. Place chile mixture in a blender or food processor; process until smooth.
Cook cumin seeds in a large Dutch oven over medium heat for 1 minute or until toasted and fragrant. Place in a spice or coffee grinder; process until finely ground.
Heat oil in Dutch oven over medium-high heat. Add pork; cook 5 minutes, browning on all sides. Remove pork from pan. Reduce heat to medium. Add onion and garlic to pan; cook 8 minutes or until onion is browned, stirring frequently. Stir in pork, pureed chiles, toasted ground cumin, broth, sugar, salt, and hominy; bring to a simmer. Cook 30 minutes or until pork is tender. Spoon 1 2/3 cups posole into each of 6 bowls; top each serving with 1 tablespoon radishes, 1 tablespoon green onions, and 1 tablespoon cilantro. Serve with lime slices.