Bring 4 cups of water, the 16 oz (454 g) pork, quartered onion, smashed garlic, thyme, and bay leaves to a simmer in a large saucepan over medium-high heat, skimming off any foam that rises to the surface.

Reduce heat to medium-low, partially cover, and cook until the meat is tender, about 1 hour and 15 to 30 minutes.

Heat coconut oil in a nonstick skillet over medium heat until shimmering.

Add the 8 oz (227 g) mushrooms and cook until they release their liquid.

Continue to cook until well-cooked, tender, and slightly browned.

Set aside.

Place 1.5 cups of water and the 7 oz (198 g) turnips in a pot over medium-high heat and cover.

Bring to a boil, then reduce to medium-low heat and simmer until crisp and tender.

Drain and set aside.

Meanwhile, adjust oven racks to the upper-middle and lower-middle positions and preheat the oven to 450° F.

Spread the 12 tortillas over two rimmed baking sheets, grease both sides with coconut oil, and bake until brown and crisp, 8 to 10 minutes, switching and rotating the sheets halfway through baking.

Drain the pork, reserving 1 cup of the cooking liquid.

Remove and discard the onion, garlic, thyme, and bay leaves. Return the pork to the saucepan and, using a potato masher, mash until the pork is shredded into rough 0.5-inch pieces.

Heat coconut oil in a 12-inch nonstick skillet over medium-high heat until just smoking.

Add the shredded pork and the chopped onion and cook until the pork is well-browned and crisp, 5 to 7 minutes.

Stir in the minced garlic and 0.5 tsp dried oregano and cook until fragrant, about 30 seconds.

Stir in the reserved 1 cup pork cooking liquid, the 15-oz (425-g) can of tomato sauce, the cooked mushrooms, the cooked turnips, and the 1 tbsp minced chipotle.

Bring to a simmer and cook until almost all the liquid has evaporated, 5 to 7 minutes; season with sea salt and pepper to taste. Spoon the pork mixture onto the tostadas and serve.