Combine flour, salt, and pepper in a resealable bag; add stew meat and shake until meat is coated.

Heat oil in a Dutch oven or large heavy-bottomed pot over medium heat. Add coated meat and loose flour from the bag to the hot oil; cook and stir until browned on all sides, 5 to 10 minutes.

Mix tomatoes, onions, tomato sauce, water, Worcestershire sauce, garlic, paprika, sugar, and bay leaves into stew meat mixture; bring to a boil. Reduce heat to low and simmer stew, scraping bottom of Dutch oven every 30 minutes, about 1 hour.

Stir carrots into stew and cook for 30 minutes more. Add potatoes to stew and cook until potatoes are tender, about 30 minutes. Add mushrooms to stew and cook until tender, about 15 minutes.