The polynomial ring ℤ[x₁₁, …, x₃₃] has a basis called the dual canonical basis whose quantization facilitates the study of representations of the quantum group U_q(sl₃(ℂ)). On the other hand, ℤ[x₁₁, …, x₃₃] inherits a basis from the cluster monomial basis of a geometric model of the type D₄ cluster algebra. We prove that these two bases are equal. This extends work of Skandera and proves a conjecture of Fomin and Zelevinsky.