Cut ribs apart between bones. Rinse and put in a 6- to 8-qt pot. Add 2 1/2 qts water, cover and bring to a boil. Skim and discard any foam.
Add mushroom soy sauce, rice wine, five-spice powder, star anise and ginger. Return to a boil, then reduce heat and simmer covered, until meat is tender (45 minutes to 1 hour).
Strain rib mixture through a colander set over a large bowl. Set bowl of braising liquid in a larger bowl of ice water. Let ribs and seasonings cool in colander, then return to liquid and chill covered, at least 4 and up to 12 hours.
Meanwhile, in a small bowl, stir together sugar, reduced-sodium soy sauce, vinegar, and fish sauce until sugar dissolves. Set aside.
Preheat oven to 250 degrees F and set a rimmed baking sheet in it. Pour 1 1/2 in oil into a 5- to 6-qt pot. Insert a deep-fry thermometer and heat oil over high heat to 350 degrees F.
Meanwhile, skim and discard fat from bowl of ribs. Transfer ribs to a colander and discard seasonings. Working with one-quarter of ribs at a time, toss in a medium bowl with 1/4 cup cornstarch.
Gently add ribs to oil. Cook until bones are browned and meat is crisp, 2 to 3 minutes. Transfer ribs to baking sheet in oven.
Heat a wok or 12-in frying pan over medium heat. Add 1 tbsp of remaining 1/4 cup oil. When it's hot, stir in one-quarter of garlic. Cook about 15 seconds, then add one-quarter each of ribs and sauce. Cook, stirring with tongs (or a wide metal spatula, if using a frying pan), until sauce thickens and coats ribs well (1 to 2 minutes). Stir in one-quarter of basil leaves; cook and toss ribs until basil turns bright green, about 15 seconds. Transfer to a platter. Scrape out excess sauce from wok and spoon over ribs.
Cook remaining ribs the same way, adding a splash of water and reducing heat if pan starts to scorch.