Let A be a local ring, and let I1,…,Ir ⊂ A be ideals of positive height. In this article we compare the Cohen–Macaulay property of the multi–Rees algebra RA(I1,…,Ir) to that of the usual Rees algebra RA(I1 ··· Ir) of the product I1 ··· Ir. In particular, when the analytic spread of I1 ··· Ir is small, this leads to necessary and sufficient conditions for the Cohen–Macaulayness of RA(I1,…,Ir). We apply our results to the theory of joint reductions and mixed multiplicities.