Though the job is spread over three days, making this pickle is actually quite easy and well worth the effort.
Wash tomatoes thoroughly and cut away any bruises or bad spots. Cut into 1/4-inch-thick slices.
Combine 2 gallons water and the pickling lime in a large nonreactive bowl and add tomatoes. Let soak for 24 hours.
Drain tomatoes and cover with fresh water. Soak for 4 hours, changing water every half hour. Rinse and drain well.
In a large nonreactive pot, combine sugar and vinegar and bring to a boil. Fold an 8-by-16-inch piece of cheesecloth in half to make a square; rinse with water and squeeze dry. Place spices and ginger on the cloth; tie closed with one end of a 12-inch piece of cotton string. Tie a loop in the other end and slip it over the handle of a wooden spoon. Suspend spice bag in the syrup by placing the spoon across the top of the pot. Remove syrup from heat, add tomatoes, and let sit overnight, covered with a clean dish towel.
Bring the tomatoes to a simmer and cook, pushing them into the syrup occasionally, until translucent, about 35 minutes.
Meanwhile, sterilize 6 pint canning jars and lids. Wash jars, lids, and screw bands in hot, soapy water; rinse well. Place jars upright on a wire rack in a large pot, fill with hot water until jars are submerged, and bring to a boil. Boil for fifteen minutes. Turn off heat; leave jars in water. Sterilize lids according to manufacturer's instructions.
Using stainless-steel tongs, layer hot tomatoes in hot sterilized pint jars, leaving 3/4 inch of space beneath the rim. Pour hot syrup over tomatoes, covering them by 1/4 inch, leaving 1/2 inch of space beneath the rim. Slide a clean plastic chopstick or wooden skewer along the inside of each jar to release any air bubbles. Wipe the mouth of the jar with a clean, damp cloth. Place hot lid on jar; screw on band firmly without forcing.
Process jars in a water bath. Place a wire rack in the bottom of a large pot and fill pot with hot water. Using a jar lifter, place the jars on the rack. Add enough hot water to cover jars by two inches, and bring to a boil. Boil for 10 minutes. Remove jars from water bath with jar lifter; let stand on clean dish towels for 24 hours. As the pickles cool, a vacuum forms inside the jar, sealing it. Check for the slight indentation in the lids that indicates a vacuum seal. Screw bands can now be removed, but leave them on to protect the seal if you're going to transport the jars. Jars that don't seal properly or that leak during processing should be stored in the refrigerator and the pickles consumed within a week. Tomatoes can be served when cool but are more flavorful when allowed to mellow in a cool, dry place for 2 to 3 weeks.