Slow-Cooker Bean and Barley Soup

1 cup dried multibean mix or Great Northern beans, picked over and rinsed
6 cups water
1 (14-ounce) can whole tomatoes, with juice
3 cloves garlic, smashed
2 ribs celery, chopped
2 medium carrots, chopped
1/2 medium onion, chopped
1/2 cup pearl barley
1 bay leaf
1 1/2 tablespoons kosher salt, plus additional for seasoning
2 teaspoons dried Italian herb blend
Freshly ground black pepper
1/2 ounce dried porcini mushrooms, optional
3 cups cleaned baby spinach leaves (about 3 ounces)
1 cup freshly grated Parmesan
1 tablespoon balsamic vinegar
Extra-virgin olive oil

Put beans, water, tomatoes and their juices, garlic, celery, carrots, onion, barley, bay leaf, 1 1/2 tablespoons salt, herb blend, pepper, and porcini mushrooms (if using) in a slow cooker; cover and cook on LOW until the beans are quite tender and the soup is thick, about 8 hours.
Stir in the spinach, cheese, and vinegar, cover, and let the soup cook until the spinach wilts, about 5 minutes. Taste and season with salt and black pepper, to taste.

Ladle the soup into warmed bowls and drizzle each serving with olive oil.