Heat the oil in a heavy pot large enough to hold everything. Brown the pork, turning now and then, about 15 minutes. Take the pork out and put it on a plate. Brown the veal, about 10-15 minutes, turning periodically, then take it out and add it to the plate. Brown the sausage, 10-15 minutes, then take it out and add to the plate.
Drain the excess fat, leaving about 1 Tbs in the pot. Cook the garlic one minute, flattening and turning now and then to flavor the oil. Take the garlic out and discard it.
Add the tomato paste, tomatoes, water, salt, pepper, and basil leaves. Stir until the paste is mixed in well. Add the meats back in. Bring to a simmer, partially cover, and cook for 2 hours or so. Stir periodically.
Add the meatballs and cook another half an hour. If the sauce becomes too thick, add some water. If it's not thick enough, take the cover off the last half an hour.