Preheat your oven to 350°F (175°C). In a large mixing bowl, combine the white cake mix, eggs, water, oil, and vanilla extract. Mix according to the package instructions until the batter is smooth and well combined.
Pour the batter into a greased 9x13 inch baking pan. Bake in the preheated oven for 25-30 minutes or until a toothpick inserted in the center comes out clean. Once baked, let the cake cool in the pan for about 10 minutes before transferring it to a wire rack to cool completely.
While the cake is cooling, prepare the pudding. In separate bowls, combine the vanilla pudding mix and banana pudding mix with 4 cups of milk. Whisk each until smooth and thickened.
Once the cake has cooled completely, use a toothpick or skewer to poke holes all over the top of the cake. Make sure to poke deep enough for the pudding mixture to soak in, but be gentle to avoid breaking the cake.
Pour the condensed milk evenly over the poked cake, allowing it to seep into the holes. Then, pour the prepared pudding mixture over the top, spreading it out evenly.
In a microwave-safe bowl, melt the white chocolate chips. Once melted, let them cool slightly before folding in the whipped topping until smooth and creamy.
Spread the whipped topping mixture over the pudding layer. Divide the whipped topping into three bowls. Use the gel food coloring to tint one bowl green, another purple, and the last yellow.
Using a spatula or spoon, dollop the different colored whipped toppings onto the cake in a random pattern. Use a knife or spatula to swirl the colors together slightly, creating a marbled effect. Finally, sprinkle the green, yellow, and purple sprinkles generously over the top.
Cover the cake and refrigerate for at least 2 hours, allowing the flavors to meld and the cake to set. When ready to serve, slice into squares and enjoy!