Combine the tomato paste, kimchi, kimchi brine, gochujang, honey, cider vinegar, onion powder, garlic powder, smoked paprika, and allspice in food processor or blender. Process until smooth.
Transfer to a jar, seal tightly and allow to ferment on the counter for up to 72 hours. Stir the ketchup once or twice a day during the fermentation period so that flavors develop evenly. Then transfer the to the fridge and store for up to 3 months.