Pulse the onion, celery and carrots in a food processor until they are pea-size pieces.
Melt the butter in a large pot over medium-high heat. Add the chopped onion, celery and carrots and the allspice and cook, stirring occasionally, until the vegetables are slightly softened, about 5 minutes.
Add the tomatoes to the food processor and pulse until finely chopped. Transfer the tomatoes to the pot and add the beef broth, cabbage, potatoes, barley and 4 cups water; cover and bring to a boil. Uncover, reduce the heat to medium low and simmer until the potatoes and barley are tender, about 20 minutes.
Stir in the corned beef and season with salt and pepper.
Courtesy of Food Network Magazine, Photograph by Antonis Achilleos