For the tomato base, grate the tomatoes using the coarse side of a box grater into a mixing bowl. Add the rest of the ingredients to the bowl and let sit for 3 hours. Reserve.
Combine all ingredients for the horseradish in a blender. Purée until very smooth. Add more water if necessary; it should be just covered with liquid. Taste for seasoning—it should taste very strong.
Mix the cocktail ingredients together and pour over ice. Serve with garnishes.