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.