We study the coupling of pairs of reverse plane partitions of the same shape by assigning a certain local interaction between the reverse plane partitions. We show that they are in bijection with a certain Yang-Baxter integrable colored vertex model. By utilizing the Yang-Baxter equation for this colored vertex model, we are able to compute the generating function for the interacting pairs of reverse plane partitions. We also give a bijection between the coupled pairs of reverse plane partitions with the interaction strength set to zero and a single reverse plane partition of the same shape.