Using the highest sauté or browning function, heat oil in an electric pressure cooker until shimmering. Add onions, Swiss chard stems, and salt, cooking until softened (3–5 minutes). Stir in garlic, 4 teaspoons of ginger, serrano chile, tomato paste, turmeric, and garam masala. Cook until fragrant (about 1 minute).
Stir in broth and 1 can of coconut milk, scraping up browned bits. Add wild rice. Lock the lid in place and close the pressure release valve. Cook on high pressure for 30 minutes. Turn off the pressure cooker and quick-release the pressure. Carefully remove the lid, allowing steam to escape.
Stir in the remaining can of coconut milk and Swiss chard leaves. Using the highest sauté or browning function, cook until chard leaves are wilted (about 5 minutes). Turn off the pressure cooker. Stir in the remaining 2 teaspoons of ginger, chopped tomato, and let sit until heated through (about 2 minutes). Stir in cilantro, adjust seasoning with salt and pepper, and serve with lime wedges.