Preheat the oven to 325°F (160°C). Line a 9×13-inch baking dish with parchment paper, leaving an overhang to lift the bars out later.
In a bowl, combine the graham cracker crumbs and gingersnap cookies. Add the melted butter, honey, and mix until the crumbs are evenly moistened. Press the crust mixture into the bottom of the prepared baking dish and bake for 10 minutes. Set aside to cool.
In a large mixing bowl, beat the cream cheese and sugar until smooth and creamy. Add the vanilla extract and beat until well combined. Gradually add the eggs one at a time, beating well after each addition. Mix in the Greek yogurt until smooth.
Pour the cheesecake mixture over the cooled crust and smooth the top with a spatula. In a small bowl, mix the raspberry jam with the lemon juice to create a smooth, spreadable consistency. Dollop spoonfuls of the raspberry mixture in stripes or swirls over the cheesecake. Use a knife or toothpick to gently create a swirl pattern in the cheesecake mixture.
Bake the bars for 25-30 minutes or until the edges are lightly golden and the center is set. Allow to cool completely at room temperature, then refrigerate for at least 4 hours or until firm.
Once chilled, lift the bars from the pan using the parchment paper, and cut into 16 equal-sized bars.
Garnish with fresh raspberries if desired, and serve.