In a large skillet over medium-high heat, heat oil. Add onion, ginger, chile, garlic, and ¼ teaspoon salt. Cook, stirring frequently, until onion is translucent and ginger and garlic are fragrant, 3 to 5 minutes.
Add tomato paste, garam masala, paprika, and turmeric and cook, stirring, until fragrant, about 1 minute. Add coconut milk, brown sugar, and ¼ teaspoon salt; bring to a simmer. Add shrimp and cook, tossing shrimp a few times, until just about cooked through, 3 to 5 minutes. Add peas and spinach and cook, tossing, until spinach is wilted, peas are warmed, and shrimp is cooked through, about 2 minutes more. Stir in lime juice; season with more salt, if needed.
Divide rice among bowls. Top with shrimp mixture. Serve with lime wedges alongside.
