Step 1

Pulse garlic in a food processor until finely chopped. Add chiles, salt, and sugar and pulse until chile pieces are no bigger than ½". Transfer mixture to jar and press down so chiles are slightly submerged in their own liquid. Cover with cheesecloth and fasten with kitchen twine or a rubber band. Let sit in a cool place away from direct sunlight at least 2 days and up to 5 days. The liquid will develop a slightly sour smell and there should be small bubbles scattered throughout. The longer you let it ferment, the more active it will become, but don’t go longer than 5 days.

Step 2

Heat oil, achiote seeds, and paprika in a small saucepan over medium-high. As soon as small bubbles appear around the edges of the pan (2–3 minutes), let cook another 30 seconds; let cool. Strain oil through a fine-mesh sieve into a heatproof bowl; discard solids.

Step 3

Blend achiote oil, vinegar, and two-thirds of chile mixture in a blender until smooth. Transfer to an airtight container and stir in remaining chile mixture.

Step 4

Do Ahead: Hot sauce can be made 2 weeks ahead; cover and chill.