Set the Instant Pot to Saute, Normal Temperature. Once the pot is hot, add the olive oil.
Add the mushrooms. Saute the mushrooms so they release water and begin to brown.
Add the carrots through the garlic (this includes: carrots, squash, zucchini, onion, and garlic), and saute the veggies to soften slightly.
Press the Cancel button, and add the can of tomatoes (undrained--you want the juices in the can). Make sure to scrape up any browned bits on the bottom of the pot, or it won’t come to pressure.
Stir in the barley and all the seasonings. Add the broth.
Lock the lid and set to Sealing. Set to Manual, High Pressure, for 7 minutes.
Carefully perform a quick release (use a hot pad holder or towel) when the time is up and before opening the lid.
Remove the bay leaves. Taste the soup and add more salt, if needed. Serve warm.