Preheat oven to 325°F (165°C). Line a 9x13-inch pan with parchment paper, leaving overhang on the sides.
Pulse the Oreo cookies in a food processor until fine crumbs form. Add melted butter and pulse to combine. Press firmly and evenly into the bottom of the pan. Bake for 10 minutes, then set aside to cool slightly.
In a large bowl, beat the softened cream cheese and sugar until smooth and fluffy (about 2–3 minutes). Add sour cream and vanilla extract and mix until combined.
Add eggs one at a time, mixing just until incorporated. Add the flour and mix on low until smooth.
Divide the cheesecake mixture into 3 bowls. Leave one bowl white. To the 2nd bowl, add a generous amount of red gel food coloring and stir to combine. To the 3rd bowl, add a generous amount of blue gel food coloring.
Drop spoonfuls of the colored batters on top, dividing the colors into sections, and gently swirl with a skewer or knife.
Place the baking pan inside a larger roasting pan. Pour hot water into the roasting pan so it reaches halfway up the sides of the cheesecake pan.
Bake for 35–40 minutes, or until the center is just set and slightly jiggly. Turn off the oven, crack the door, and let the bars rest inside for 10 minutes.
Carefully remove the cheesecake from the water bath. Let it cool at room temperature, then refrigerate for at least 4 hours (or overnight) before slicing into bars.