Puree strawberries in a blender until smooth. Divide strawberry puree evenly among 4 tall glasses.
Place ice in blender; add mango and next 4 ingredients. Process until smooth. Pour mango-papaya mixture evenly over strawberry puree in glasses.
"Warm" Ice: Slightly melted ice blends a better drink, Ernie says. He suggests letting the ice sit out at room temperature for about 30 minutes before using. "It should be wet to the touch without sticking to your fingers," he explains.