For the pineapple-pomegranate relish: Combine the pineapple, cilantro, pomegranate, chives and lime juice in a bowl. Add honey if needed. Season the relish with salt and pepper and let sit at room temperature at least 30 minutes to allow the flavors to meld.
For the pork: Preheat the oven to 325 degrees F.
Put the pork into a large bowl and sprinkle with the ancho, New Mexican and arbol chile powders and the salt, tossing to coat evenly.
Heat the oil in a large roasting pan or braising pan over high heat until the oil begins to shimmer. Add the pork and sear until golden brown on all sides. Add the pineapple, garlic and onions and cook for 2 minutes. Add the chicken stock, pineapple juice and pomegranate juice and bring to a boil.
Cover the pan tightly with aluminum foil and put in the oven. Braise until the pork is fork-tender, about 1 1/2 hours. Remove the pork to a bowl and let cool slightly, then coarsely shred with two forks.
Mix together the pomegranate molasses, honey and 1/2 cup of the braising liquid in a large saute pan and bring to a boil over high heat. Add the shredded pork and toss to coat.
For the puffy tacos: Combine the masa mix, salt and 2 1/4 cups warm water in a large bowl and mix until a smooth dough forms. Pull off pieces of dough and roll them into balls about the size of a ping pong ball.
Meanwhile, pour 2 inches of oil into a large deep pot and heat until a deep-fry thermometer inserted in the oil registers 350 degrees F.
Cut a quart-size resealable plastic bag open down both sides to form a rectangle. Use the bag to line the tortilla press as you shape the tortillas so they do not stick: Lay one side of the plastic over the bottom of the press. Place a ball of dough in the center and fold the other side of the plastic over the dough. Shut the top and press firmly down on the dough to shape the tortilla. Continue with the remaining dough balls.
Drop a tortilla into the hot oil and, using a metal spatula, repeatedly douse the tortilla with the hot oil until it begins to puff up. Flip it over and, using the spatula, make an indentation in the center of the tortilla to form a taco shape. Transfer to paper towels to drain and sprinkle with salt.
Fill the puffy taco shells with some of the pork, avocado slices, cheese, pineapple-pomegranate relish, cilantro and radish slices and serve.