Rinse the split peas and pick out any debris. Combine them with the water or broth in a heavy-bottomed pot. Bring to a boil, then reduce to a low simmer, covered, for about 1 hour until the peas are soft. Add water as needed to keep it from getting too thick or scorching.
(Optional) For a smooth soup, puree with an immersion blender or in a blender, then return it to the pot. Keep it thin enough to pour.
Peel and slice the carrots, and peel and chop the onion. Add them to the pot along with the ham, bay leaf, and allspice.
Prepare and preheat your pressure canner for hot pack canning.
Simmer the soup for 30 more minutes until the vegetables are cooked, adding water as needed to keep it pourable.
Adjust the seasonings and salt to taste, then remove the bay leaf.
Ladle the hot soup into prepared jars, leaving 1 inch of headspace. Remove air bubbles, wipe the rims, and apply lids and bands to fingertip tight.
Load the jars into the canner, lock the lid, and vent steam for 10 minutes. Bring to 10 lbs pressure for a weighted gauge canner or 11 lbs for a dial gauge, adjusting for altitude.
Process pints for 75 minutes or quarts for 90 minutes.
Turn off the heat and let the canner return to zero pressure naturally. Cool the jars 12 hours, then check the seals before storing.