Puree the strawberries and slightly less than half of the raspberries in a food processor until smooth. Set aside the whole raspberries. Strain the puree through a sieve to eliminate seeds, pushing the puree through with a rubber spatula or the back of a spoon.
Flavor the puree to taste with sugar, if needed, plus lemon juice. Stir in the reserved whole berries. Cover and chill the sauce.