Place all of the prepared and drained vegetables in separate bowls and reserve their juices.
In a large saucepan, combine ketchup, Worcestershire, horseradish, tomato sauce, tarragon vinegar, brown sugar, garlic, lemon juice, and cayenne. Bring to a simmer over medium heat. Season, to taste, with salt.
Bring a large pot of water to a boil and salt it generously. Blanch the cauliflower for about 2 minutes; it should still be crisp. With a skimmer, transfer the florets to a tray to cool quickly. Repeat with the carrots and then with the celery, cooking them just long enough to remove their raw taste; they will be cooked more later. Add the white wine vinegar to the water and blanch the mushrooms for 1 minute (the vinegar will keep the mushrooms white). Drain and spread the mushrooms on a tray to cool quickly.
Add the cauliflower, carrots, celery, and mushrooms to the sauce. Add the juices from the canned and jarred vegetables. Add the pepperoncini, olives, pickles, onions, and artichoke hearts and toss well.
In sterilized canning jars, place 1 teaspoon tarragon vinegar (this will guarantee freshness). Load the sauced vegetables into the jar, using a wide-mouthed funnel to avoid a mess.
In one of the large pans that you have used, add olive oil, tuna, and anchovies and mix well. Top each jar with some of the tuna mixture. Cover and refrigerate, or allow to cool on the counter before serving. This will keep in the refrigerator for up to 2 weeks.
Properly handled sterilized equipment will keep canned foods in good condition for one year. Making sure hands, equipment and surfaces in your canning area are clean is the first step in canning. Tips: Jars should be made from glass and free of any chips or cracks. Preserving or canning jars are topped with glass, plastic or metal lids that have a rubberlike seal. Two-piece metal lids are most common. To prepare jars before filling: Wash jars with hot, soapy water, rinse them well and arrange them open-side up, without touching, on a tray. To sterilize jars, boil them in a large saucepan, covered with water, for 10 minutes. Jars have to be sterilized only if the food to be preserved will be processed for less than 10 minutes in a boiling-water bath or pressure canner. To sterilize jars, boil them in a large saucepan, covered with water, for 10 minutes. Follow manufacturer's instructions for cleaning and preparing lids and bands. Use tongs or jar lifters to remove hot sterilized jars from the boiling water. Be sure the tongs are sterilized too: Dip the tong ends in boiling water for a few minutes before using them. All items used in the process of making jams, jellies, preserves and pickles must be clean, including any towels and especially your hands. After the jars are prepared, you can preserve the food. It is important to follow any canning and processing instructions included in the recipe and refer to USDA guidelines about the sterilization of canned products. Find Information information on canning can be found at the National Center for Home Food Preservation website: http://nchfp.uga.edu/.