Place smoked turkey wings in a large stockpot. Cover with water. Bring to a boil. Reduce to a simmer over medium heat. Add green beans and bay leaves. Bring back to a boil; reduce heat to medium and simmer until green beans soften, 1 to 1 1/2 hour.
Remove turkey wings from the stockpot and cool until easily handled, about 15 minutes.
Stir potatoes into the stockpot. Shred turkey meat and stir in. Season with garlic salt and black pepper. Cook, uncovered, stirring occasionally, until potatoes are nearly falling apart, about 25 minutes. Discard bay leaves before serving.