Bring a large pot of salted water to a boil. Add shrimp; cook just until curled and pink, about 1 minute. With a slotted spoon, transfer shrimp to a large bowl of ice water. Let stand 5 minutes. Drain.
While shrimp are cooling, bring water back to a boil. Cook noodles according to package directions, stirring occasionally and skimming any scum from surface. Drain; rinse under cold water until cool. Drain well and transfer to a large bowl.
In a small bowl, whisk together mirin, lime juice, orange juice, soy sauce, sesa- me oil, ginger, and salt. Add 1/2 cup dressing to noodles; toss to coat. Add shrimp, green onions, corn, radishes, cucumber, and chopped basil and toss to combine.
Divide noodles among 4 shallow bowls. Top with basil leaves and drizzle bowls with remaining dressing. Serve with lime wedges.