Tomato is not a traditional ingredient of corned beef and cabbage soup, but I added some for a bit of tartness to contrast with the salty beef and the natural sweetness of white cabbage.
Put the beef in the slow cooker and cover with cold water.
Add the onion halves and garlic cloves.
Cook on HIGH for two hours then switch to LOW and cook for another three hours.
Scoop out the beef.
Strain the broth and pour back into the slow cooker.
Add the tomatoes, potatoes and cabbage to the broth.
Cook on HIGH for another 40 to 50 minutes or until the vegetables are soft.
Taste. Add salt, if needed (chances are you won't need to because the cooking liquid is already perfectly seasoned after the corned beef has cooked in it).
Cut the corned beef into cubes and add to the vegetables in the cooker.
Continue cooking for another 10 minutes or until the soup is bubbly.
Ladle the corned beef and cabbage soup into bowls, and serve immediately.