You need not case this sausage. I often make it loose, as most Mexican recipes call for uncased chorizo anyway.
Take out some hog casings and set in a bowl of very warm water. Mix the tequila and ice water with the achiote paste and chill it in the fridge.
Grind the meat mixture through your meat grinder (you can use a food processor in a pinch, but you will not get a fine texture), using the fine die, about 4.5 mm. If your room is warm, set the bowl for the ground meat into another bowl of ice to keep it cold. Make sure the meat mixture is very cold before moving on to the next step: You want it between 27°F and 35°F.
Hang the sausages in a cool place for up to overnight (the colder it is, the longer you can hang them). If it is warm out, hang for just one hour. Once they have dried a bit, put in the fridge until needed. They will keep for at least a week in the fridge. If you are freezing the sausages, wait a day before doing so. This will tighten up the sausages and help them keep their shape in the deep-freeze.