Wash the tomatoes. Dip them in boiling water for 30-40 seconds, or until the skins split and pucker. Then, dip in cold water or a bucket of ice and remove the skins.
Core and quarter the tomatoes. Then, boil them for 20 minutes in a large, uncovered saucepan. Run the tomatoes through a sieve or food mill.
Saute the onions, celery, peppers, mushrooms, and garlic in olive oil (or vegetable oil) until they are tender.
Combine the vegetables with the tomatoes, then add the rest of the ingredients. Bring everything to a boil, then simmer uncovered until the sauce is thick enough for serving. Stir often to prevent burning. The sauce should reduce by about half.
While the sauce cooks, prepare a pressure canner for hot pack, along with jars, lids and rings.
Once your sauce is ready, fill hot, sterilized jars, leaving 1'' headspace.
Wipe the rims of the jars to remove food particles. Add the lids and bands to the jars, then tighten until they are fingertip tips.
Load the jars into the canner.
Turn heat to high, then allow the canner to vent steam for 10 minutes.
Add the weight, then process the jars at 11 lbs pressure for a dial gauge canner or 10 lbs pressure for a weighted gauge scanner. You will process pints for 20 minutes and quarts for 25 minutes. Adjust for altitude as needed.
When the processing time has finished, allow the canner to depressurize on its own.
Remove the jars, then allow them to cool for 12 hours. Check the seals before storing.