One 2-pound head of green cabbage, cut into 6 wedges
Mustard and horseradish, for serving
Put the corned beef in a large slow cooker. Add the garlic and 6 cups of the water, cover and cook on high for 5 hours and 30 minutes.
Add the remaining 4 cups of water and cook on low until the corned beef is tender, about 1 hour and 30 minutes longer. Transfer the corned beef to a large pan and cover with foil. Discard the garlic.
Add the carrots and potatoes to the cooker, cover and cook on high until almost tender, about 2 hours. Add the cabbage wedges and cook until tender, about 45 minutes.
On a carving board, carve the corned beef across the grain into 1/3-inch-thick slices. Keeping the slices together, return the meat to the slow cooker and cook on low until heated through, about 15 minutes. Serve the corned beef and vegetables in the cooking liquid, passing mustard and horseradish at the table.
MAKE AHEAD The corned beef and vegetables can be refrigerated in the cooking liquid overnight and reheated gently.