1. In a large heavy stockpot set over high heat, combine oil, onions, and jalapenos. Cook, stirring, until vegetables are soft, 2 to 3 minutes. Add garlic, pepper, anchovies, cloves, salt, lemons, corn syrup, cane syrup, vinegar, 4 cups water, and horseradish; bring to a boil.

2. Reduce heat to medium-low and simmer, stirring occasionally, until mixture barely coats a wooden spoon, about 6 hours. Strain through a fine mesh strainer and let cool to room temperature. Use immediately or keep refrigerated indefinitely.

3. To store sauce at room temperature for future use, place 3 clean pint jars right side up on a rack in a boiling-water canner. Fill the canner and jars with hot water, about 1 inch above the tops of jars. Boil jars over high heat for 10 minutes. Remove and drain hot sterilized jars one at a time, reserving hot water for processing filled jars. Place jars on a wire rack set over a rimmed baking sheet.

4. In a large saucepan filled with water, bring to a boil over high heat and reduce to a simmer, add clean lids and lid rings. Simmer for 10 minutes; do not boil, as this may cause problems in sealing jars. Drain lids and rings; set aside.

5. Ladle Worcestershire sauce into a jar up to the fill line. Repeat process with remaining jars. Put lids and rings on jars and tighten; do not over-tighten.

6. Reheat water in the canner until it reaches at least 180 degrees, within 10 minutes of filling the jars. Place filled jars into the canner one at a time, using a jar lifter that is securely positioned below the neck of the jar. Keep jars upright at all times.

7. Add more boiling water, if needed, so that water covers jars by at least 1 inch. Increase heat to high and cover. Once water begins boiling, heat jars for 5 minutes. Turn off heat and gently transfer jars to a wire rack set over a rimmed baking sheet and invert, spacing each jar at least 1 inch apart. Avoid placing jars on a cold surface or near a cold draft.

8. Let jars sit undisturbed until fully cooled, 12 to 24 hours. Do not tighten ring bands on the lids or push down on the center of the flat metal lid until jar has cooled completely.

9. Once jars have cooled completely, test to make sure each jar is completely sealed. Press down on the middle of the lid with a finger. If lid springs up when finger is released, the jar is unsealed. Store sealed jars in a cool place for at least 2 weeks. If any of the jars are unsealed, store in the refrigerator and use within several days. Always refrigerate sauce after opening.