In a blender, combine the strawberries, 2 tablespoons sugar and 1½ tablespoons lime juice. Puree, scraping down the sides as needed. Pour through a fine-mesh sieve into a bowl, pressing on the solids; discard the solids. Divide the puree among six 4-ounce or eight 3-ounce popsicle molds. Freeze until firm, about 1 hour.
Combine yogurt, cream and 2 tablespoons each sugar and lime juice. Remove the molds from the freezer and top with the yogurt mixture; insert popsicle sticks. Freeze until firm, about 1 hour.
In a clean blender, puree the blueberries, remaining 2 tablespoons sugar, and remaining 1½ tablespoons lime juice. Pour through a clean sieve into a bowl, pressing on the solids; discard the solids. Remove the molds from the freezer and top with the blueberry puree. Freeze until solid, 6 hours more (or up to one week). Just before serving, briefly run the molds under hot water to release the popsicles.