1. Process first 9 ingredients, in batches, in a blender until smooth. Stir in 1/2 cup chopped cilantro. Chill at least 1 hour or up to 1 day.
2. Slice lobster tails into medallions, and split each claw into 2 pieces by cutting across the flat side.
3. Combine crema, lime juice, and lemon juice in a small bowl.
4. Divide gazpacho among 6 bowls. Top evenly with lobster, crema mixture, and remaining 1/4 cup cilantro.