Preheat the oven to 350 degrees F (175 degrees C).

Blend flour, butter, and sugar together in a food processor until crumbly. Pour in water; mix until crust dough holds together. Press into the bottom and sides of a 9-inch tart pan. Poke holes in dough with a fork to eliminate air bubbles.

Bake in the preheated oven until golden brown, about 20 minutes. Remove from oven; increase oven temperature to 375 degrees F (190 degrees C).

Chop chocolate and place in a microwave-safe bowl. Microwave in 15-second intervals, stirring occasionally, until melted; spread over baked crust.

Beat cream cheese, sugar, flour, vanilla extract, and egg together in a bowl using an electric mixer until smooth; spread filling over chocolate layer.

Arrange berries tightly and evenly over cream cheese filling. Mix raspberry syrup and blackberry schnapps together in a small bowl; brush over berries using a pastry brush.

Bake in the preheated oven until berries are reduced and filling begins to set, about 35 minutes. Remove from oven; filling will set completely as it cools.