Heat the vegetable oil in a large pot over medium heat. Stir in the onion, garlic, and ginger; cook and stir until the onion has softened and turned translucent, about 5 minutes. Sprinkle in the allspice, turmeric, coriander, and thyme; cook 1 minute longer.
Pour in the vegetable stock and sweet potatoes and bring to a boil; reduce heat to medium-low; cover and simmer 5 minutes. Add the kale and okra; simmer 5 minutes. Stir in the coconut milk, tomato, black-eye peas, and lime juice. Simmer about 5 minutes longer until the vegetables are tender.