Chop up vegetables (onion, carrots, celery, and mushrooms.)
Add 2.5 quarts of water to your Instant Pot along with the tomato paste, the chopped veggies (ie, onion, celery, carrots, and mushrooms), the barley, and the seasonings (ie, salt, pepper, thyme, bay leaf).
Give it a quick stir with a large spoon, and then seal the Instant Pot and set for manual pressure, 20 minutes. No need to distribute the tomato paste, it'll dissolve/distribute during cooking.
Use the quick release feature, when the soup in finished cooking to release the steam from the Instant Pot.
Remove bay leaf and top with grated cheese if desired.