Sterilize jars, rings, and lids according to the instructions in Home Canning (Boiling Water Method). Keep the jars in the 12-16 quart pot and the lids and bands in the 3-quart pot with hot, not boiling, water. (Optional: My sister does the dishwasher method and turns on the heated dry cycle to get the jars good and hot just before adding the ingredients.)
Rinse the tomatoes, peppers, and dill. Quarter the green tomatoes, removing any bad spots. Place the peppers and garlic in the bottom of the jars, then add tomatoes up to about ½ inch (1 cm) from rim of the jar. Add the olive oil, pickling spice, alum (optional), and fresh dill.
In the 6-quart pot, cook vinegar, water, and coarse salt over high heat until it comes to a strong rolling boil. The jars have to seal based solely on the heat of this mixture, so keep it good and hot the whole time.
Portion the vinegar solution into the jars, adding enough to cover the tomatoes and seasonings, but still leaving ½ inch (1 cm) of headspace. If the tomatoes were packed very tight, run a plastic spatula inside the jar to get rid of any air bubbles.
Tighten lids and wipe off edges, and leave out on the counter for about 1 hour. Listen for the popping sound to make sure they seal. Label the jars when cool.
To give the pickles time to develop flavor, set aside for at least two weeks before eating. Store in a cool, dark, dry place and use within 3 months.