Add the blackberries to a small saucepan and cook on medium-high heat until they release their juice and start breaking down. Push the mixture through a sieve to remove seeds and skin, then return the juice to the pan and simmer until it thickens into a syrupy consistency. Let it cool completely before using, since a warm reduction will melt into the filling instead of swirling on top.
Preheat your oven to 355ºF (180ºC) and line your pan with parchment paper. Mix the crushed biscuits with melted butter until it looks like wet sand, then press it firmly into an even layer using the bottom of a glass. Bake for 10 minutes, then set it aside to cool slightly while you make the filling.
Lower the oven to 285ºF (140ºC). In a large bowl, mix the softened cream cheese and yogurt together using a whisk, stirring gently rather than beating air into it. Add the sugar and cornstarch, then the eggs one at a time, mixing just until each is incorporated and smooth.
Stir in the lemon juice, vanilla, and lemon zest until the filling is silky and well combined. Take out about 1 1/2 tablespoons of this filling and mix it into the cooled blackberry reduction this keeps the swirl sitting on top instead of sinking. Pour the rest of the filling over the cooled crust and smooth it into an even layer.
Spoon the blackberry mixture over the cheesecake filling in small dollops, then drag a toothpick through it to create swirls. Go easy here, since over-swirling blends the colors together instead of keeping those pretty ribbons distinct. Scatter the fresh blackberries evenly across the top before it goes into the oven.
Bake at 285ºF (140ºC) for 30-35 minutes, until the edges look set and lightly golden but the center still has a gentle wobble. Turn the oven off and leave the door slightly open, letting the bars cool gradually inside for about an hour to prevent cracking.
Once the bars reach room temperature, transfer them to the fridge for at least an hour, though overnight gives you the best flavor and texture. Chilling firms everything up so you get clean, neat slices when it's time to cut and serve.