Combine the oil, onions, and jalapeños in a large, heavy-bottomed stockpot over high heat. Cook, stirring, until slightly softened, 2 to 3 minutes. Add the garlic, pepper, anchovies, cloves, salt, lemons (just put them in whole), corn syrup, cane syrup, vinegar, water, and horseradish, and bring to a boil. Reduce the heat to a simmer and let gently bubble, uncovered and stirring occasionally, until the mixture barely coats a wooden spoon, about 6 hours.

Strain through a fine-mesh strainer into an airtight container; discard the solids. Let cool to room temperature, then refrigerate. It will keep in the refrigerator for up to 1 month, or you can process it as directed (see next step) and it will keep in a cool, dark place for up to 1 year. Once opened, it must be refrigerated and will keep for a month.

Sterilize 3 pint-size jars and their metal lids according to the manufacturer’s instructions. Spoon the hot mixture into the jars, filling to within 1⁄2 inch of the rim. With a clean, damp towel, wipe the rims and fit with a hot lid. Tightly screw on the metal ring. Place, without touching, on a rack in a large, deep canning kettle or stockpot of rapidly boiling water; the water should cover the jars by 1 inch. Boil and process for 15 minutes. Using tongs, remove the jars from the water, place on a clean, dry towel, and let cool completely before storing. Test the seals and tighten the rings as needed. If the jars have properly vacuum-sealed, the lids will be concave. (If they aren’t, store the jars in the refrigerator and use within a month.) Store in a cool, dark place for at least 2 weeks before using.