In a medium saucepan, beat eggs, buttermilk, sugar, and lime juice until blended. Cook over low heat, stirring constantly, until the temperature of the mixture reaches 160 degrees. In a blender, blend berries and egg mixture at a low speed for about 30 seconds. Add 1 cup of ice cream to blender. Blend at high speed for 30 to 60 seconds. Serve cold.