In a saucepan, combine the mixed berries, 2 tablespoons of lemon juice, 50 grams of sugar, and 1 teaspoon of vanilla extract with a pinch of salt. Cook over medium heat for about 10 minutes, stirring occasionally and breaking up the berries until the mixture thickens.

Remove the berry mixture from heat and transfer it to a blender. Blend until smooth, or use an immersion blender right in the pot. Strain through a fine sieve to remove any seeds, then let the sauce cool before refrigerating.

In a food processor, pulse the graham crackers, 2 tablespoons of granulated sugar, and a pinch of salt until finely ground. Pour in the melted butter and pulse again until the mixture is cohesive.

Line a 23×23-centimetre baking pan with parchment paper. Evenly press the crust mixture into the bottom of the pan using a measuring cup or your fingers. Freeze it while you prepare the filling.

In a chilled mixing bowl, whip the cold heavy cream with a hand mixer until stiff peaks form, about 2 to 3 minutes. Start on low speed and gradually increase. Be careful not to over-beat the cream. Set it aside in the refrigerator.

In another bowl, beat together the room-temperature cream cheese and powdered sugar until smooth. Add the sour cream, 2 teaspoons of vanilla extract, 1 teaspoon of lemon juice, and a pinch of salt. Mix until everything is blended well.

Gently fold the whipped cream into the cream cheese mixture using a rubber spatula until just combined.

Spread the cheesecake filling evenly over the prepared crust in the pan. Drop spoonfuls of the reserved berry sauce over the surface and use a knife to swirl it gently, creating a marbled effect.

Cover the pan with plastic wrap and refrigerate for at least 8 hours or overnight to allow the bars to fully set before slicing.