Preheat oven to 350°F (175°C). Line a 9x13 inch baking pan with parchment paper, leaving an overhang for easy removal.

Prepare the Crust: In a medium bowl, combine the flour and 1/2 cup sugar. Cut in the cold butter using a pastry blender or your fingers until the mixture resembles coarse crumbs. Press the mixture evenly into the bottom of the prepared baking pan.

Bake the crust for 15 minutes until lightly golden. While the crust bakes, prepare the fruit topping. In a saucepan, combine the rhubarb, strawberries, 1/2 cup sugar, and cornstarch. Cook over medium heat, stirring occasionally, until the mixture has thickened slightly (about 5-7 minutes). Remove from heat and let cool slightly.

Prepare the Filling: In a large bowl, beat the softened cream cheese and 1/2 cup sugar until smooth. Beat in the eggs one at a time, then mix in the vanilla extract until just combined. Do not overmix.

Assemble and Bake: Pour the cheesecake filling evenly over the warm (or slightly cooled) crust. Dollop spoonfuls of the strawberry rhubarb topping over the cheesecake layer. Use a knife or skewer to gently swirl the fruit mixture into the filling if desired.

Bake for 30-35 minutes, or until the edges are set and the center is only slightly jiggly. Let the bars cool completely on a wire rack before chilling in the refrigerator for at least 4 hours (preferably overnight) before slicing and serving.