Brown beef, pork and onion in a 6-quart pot over medium-high heat for 8 minutes, breaking the meat into small crumbles and cooking it to 160°F. Drain excess fat.
Stir in garlic and caraway for 30 seconds, then add cabbage and cook for 3 minutes, turning it through the meat.
Add tomatoes, broth, salt, pepper and brown sugar. Bring to a simmer, cover partly and cook for 15 to 18 minutes until cabbage is tender.
Break up any clumps of cooked rice and stir the grains into the soup. Simmer for 3 minutes until heated through to 165°F.
Stir in cider vinegar and divide among 6 bowls while the cabbage ribbons and rice are still distinct.