Place split peas in a fine mesh strainer and rinse under cold running water to remove any dust.
Transfer drained peas into a heavy-based pot with a lid. Add water, onion, celery, ketchup, olive oil, salt, garlic and bay leaves. Cover and bring to the boil over medium-high heat. Reduce heat to a very slow simmer. Cook, stirring and scraping the bottom every 7 to 12 minutes, until peas have largely fallen apart, about 1 1/2 hours.
Stir carrots into the soup. Cook, covered, stirring often, until peas are completely dissolved and soup is thick, about 1 1/2 hours more. Remove and discard bay leaves. Season with pepper.