Place chiles in a medium bowl. Add enough boiling water to cover, and set a small plate or bowl on chiles to keep submerged. Let soak until softened, about 30 minutes. Drain chiles, reserving 1 cup soaking liquid.

Preheat oven to 350°F. Place chiles, sugar, lime juice, and ¼ cup reserved soaking liquid in a blender. Purée chile mixture, adding more soaking liquid as needed to form a smooth paste. Season pork shoulder generously with salt and spread paste over pork. DO AHEAD: Can be rubbed 1-2 days ahead. Cover and chill. Let come to room temperature before continuing.

Heat oil in a large heavy pot over medium heat. Add onion, garlic, bay leaves, oregano, coriander, cumin, and allspice. Cook, stirring often, until onion is soft, about 8 minutes. Add beer; bring to a boil. Add pork to pot; cover and transfer to oven.

Braise pork, basting occasionally with pan juices, until very tender, about 2 ½ hours. DO AHEAD: Can be made 2 days ahead. Let cool slightly. Chill uncovered until cold, then cover and keep chilled. Rewarm before continuing.

Transfer pork to a large platter. Let cool slightly. Pour pan juice mixture over pork; cover and keep warm. (Alternatively, shred the pork with 2 forks and pour pan juice mixture over pork.)

Meanwhile, working in batches, cook tortillas in a large heavy skillet over low heat until toasted, about 1 minute per side. Transfer to a large sheet of foil; wrap to keep warm.

Serve pork with tortillas, radishes, both salsas, pickled onions, and chopped cilantro, encouraging guests to fill and garnish tacos as desired.