Combine the yogurt and 1/4 cup sugar in a blender and blend for about 30 seconds. Taste it for sweetness, adding more to your liking. Transfer the mixture to a bowl — if you find it slightly grainy, let it chill in the refrigerator for 1 hour until the sugar is totally dissolved.
Rinse the blender, then add the blackberries, lime juice and 4 tablespoons of the sugar. Blend until puréed, taste, and add up to 2 tablespoons more sugar if needed. Force the berry purée through a fine-mesh sieve into a second bowl, pressing on the solids to extract as much as possible.
Alternating spoonfuls of the yogurt mixture and blackberry purée, fill the popsicle moulds, leaving about 1/2 inch at the top to allow for expansion as they freeze. Use a skewer or a paring knife to dip down into each pop and swirl a bit. Insert the sticks and transfer to the freezer; freeze 4 hours or more before serving.
Notes
Strain the seeds: Push the blackberry purée through a sieve so the pops are smooth, not seedy.Dissolve the sugar: If the yogurt mix tastes grainy, chill it an hour so the sugar fully dissolves for a smooth pop.Swirl, don't mix: Drag a skewer through the layers just once or twice for a marbled look.Leave headspace: Fill to 1/2 inch from the top so they don't overflow as they freeze.