Prepare the tomatoes: if the tomatoes are less than 3 inches in diameter, cut them into 1/4 inch slices. If they are greater than 3 inches in diameter, cut them in 1/4 inch slices, and then cut the slices into quarters so you will be able to fit them in the jars.
In a large bowl, combine the 3/4 cup of pickling lime with 3 quarts of cold water, stirring well to combine. Add the tomatoes, cover the bowl, and let it sit on the counter for 8 hours or overnight.
The next day, drain the tomatoes, cover again with cold water, and soak for one hour. Do this two more times, draining and covering with fresh cold water each time, until you have soaked the tomatoes in fresh water a total of three times, for an hour each time. This step is important; it removes all of the lime so the tomatoes will be acid enough to can safely. DO NOT SKIP ANY OF THE SOAKING STEPS. Drain the tomatoes and set aside.
Using a quadruple layer of cheesecloth, make a spice bag and tie the celery seeds, cloves, allspice, mustard seeds, and cinnamon up in it.
In a large pan or stock pot (at least 6 quarts), combine the vinegar, sugar, salt, spice bag, and diced ginger. Bring to a boil, then add the drained tomatoes. Return to a boil, then simmer over medium heat for 15 minutes, stirring occasionally and pressing down on the tomatoes to keep them submerged in the brine.
Stack the tomatoes in wide-mouth pint sized canning jars, placing a slice or two of onion between some of the layers in each jar. Place about a half teaspoon of the ginger pieces in each jar, discarding the remaining ginger, but reserving the syrup.
Leave 1" headspace in each jar between the tomato slices and the top of the jar, and cover the tomato slices with the hot syrup, leaving 1/2" of headspace. Use a chopstick or other thin, non-metal utensil to run around the sides of the jar to make sure there are no air bubbles left in the jar. Wipe the rims of the jars with damp paper towels to remove any syrup which got on the rims or the threads. Place the lids and the bands on the jars, just tightening the bands fingertip tight.
Process in a boiling water bath for 10 minutes, then remove and let sit, undisturbed, for at least 12 hours before checking seals. It is important to let them sit undisturbed for 12 hours because the sealing compound on the lids is still cooling and hardening, completing the seal. While the jars cool, you will hear a "plink" type sound from each jar - this is the jars completing the vacuum seal as the final air escapes the jar. After 12 hours have passed, remove the bands and check the lids - press down in the center of the lid. If you cannot push the lid down any further, the jar is sealed. If the lid "gives" a bit, and you can push it down, the jar did not seal. You can either put the band back on the jar, and reprocess it for another 10 minutes, or you can just put it in the fridge and use it within 3 months.