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1. Adjust oven rack to lower-middle position and heat oven to 300 degrees. Combine pork, 1 teaspoon salt, 1/2 teaspoon pepper, cumin, onion, bay leaves, oregano, lime juice, and water in large Dutch oven (liquid should just barely cover meat). Juice orange into medium bowl and remove any seeds (you should have about 1/3 cup juice). Add juice and spent orange halves to pot. Bring mixture to simmer over medium-high heat, stirring occasionally. Cover pot and transfer to oven; cook until meat is soft and falls apart when prodded with fork, about 2 hours, flipping pieces of meat once during cooking.
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2. Remove pot from oven and turn oven to broil. Using slotted spoon, transfer pork to bowl; remove orange halves, onion, and bay leaves from cooking liquid and discard (do not skim fat from liquid). Place pot over high heat (use caution, as handles will be very hot) and simmer liquid, stirring frequently, until thick and syrupy (heatsafe spatula should leave wide trail when dragged through glaze), 8 to 12 minutes. You should have about 1 cup reduced liquid.
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3. Using 2 forks, pull each piece of pork in half. Fold in reduced liquid; season with salt and pepper to taste. Spread pork in even layer on wire rack set inside rimmed baking sheet or on broiler pan (meat should cover almost entire surface of rack or broiler pan). Place baking sheet on lower-middle rack and broil until top of meat is well browned (but not charred) and edges are slightly crisp, 5 to 8 minutes. Using wide metal spatula, flip pieces of meat and continue to broil until top is well browned and edges are slightly crisp, 5 to 8 minutes longer. Serve immediately with warm tortillas and garnishes.
