Heat olive oil in a large pot over medium-high heat.
Sear the ostrich neck on all sides until browned. Remove and set aside.
In the same pot, sauté onions, carrots, and celery until softened.
Add garlic and tomato paste, cooking for another minute.
Pour in pomegranate juice or apple cider vinegar, scraping up any browned bits.
Add canned tomatoes, stock, bay leaves, rosemary, and thyme. Stir well.
Return the ostrich neck to the pot. Cover and simmer for 3-4 hours until tender.
Remove the meat from the bones, shred it, and return it to the sauce.
Season with salt and black pepper to taste.
Serve over pasta, polenta, or mashed potatoes.