In a saucepan, bring half the coconut milk to a gentle boil over medium-high. Cook, swirling the pan, until it thickens and forms large bubbles around the edge, about 2 minutes. Whisk in the curry paste and boil gently, stirring occasionally, 2 minutes. Stir in 1/2 cup water, the remaining coconut milk, fish sauce and brown sugar; bring to a boil. Cook, stirring constantly, 2 minutes; season with salt. Stir the edamame and corn into the curry and boil gently until heated through, 2 to 3 minutes. Stir in the pepper strips and heat through, about 1 minute. Serve over the rice and top with the basil.