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1
Place the pork shoulder in a deep pot with the half onion (whole), the 2 whole garlic cloves, salt, and enough cold water to submerge the meat by 3 cm. Bring to a boil over high heat, skim off any grey foam that rises, then reduce to a steady simmer. Poach uncovered for 1 hour 30 minutes to 2 hours. The pork is done when a fork meets no resistance at the thickest point and the meat begins to pull apart under gentle pressure.
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2
Transfer the pork to a board. Strain and reserve the poaching liquid. While the meat is still warm — not cold — shred it by hand or with two forks into consistent, moderately coarse pieces, discarding any bone and large fat deposits. The shreds should be warm enough to separate cleanly with uniform texture.
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3
Blend the tomatoes, chipotles, adobo sauce, and chopped garlic to a smooth purée. Heat the oil in a wide, heavy pan over medium-high heat. Add the thinly sliced onion and cook, stirring occasionally, until softened and beginning to colour at the edges — about 6 to 8 minutes. The onion should smell sweet and faintly caramelised.
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4
Pour the tomato-chipotle purée into the pan with the onion. Add the crumbled Mexican oregano and 120 ml of the reserved poaching liquid. Stir to combine and bring to a brisk simmer. This is the step on which the dish lives or dies: the sauce must reduce until it is thick enough to coat the back of a spoon and cling to the pork without pooling — a loose, watery sauce will collapse the taco and drown the pork's flavour. Continue reducing, stirring frequently, until this consistency is reached, approximately 10 to 15 minutes.
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5
Add the shredded pork to the pan. Fold to coat every strand in the sauce. Reduce heat to medium and continue cooking for 5 to 8 minutes, allowing the pork to absorb the sauce and any remaining moisture to evaporate. Taste and adjust salt and chipotle level. The tinga is ready when the mixture moves as a cohesive mass and leaves the pan sides clean as it is stirred.