The bacterium E. coli is widely used to produce recombinant proteins such as growth hormone and insulin. One inconvenience with E. coli cultures is the secretion of acetate through overflow metabolism. Acetate inhibits cell growth and represents a carbon diversion, which results in several negative effects on protein production. One way to overcome this problem is the use of a synthetic consortium of two different E. coli strains, one producing recombinant proteins and one reducing the acetate concentration. In this paper, we study a chemostat model of such a synthetic community where both strains are allowed to produce recombinant proteins. We give necessary and sufficient conditions for the existence of a coexistence equilibrium and show that it is unique. Based on this equilibrium, we define a multi-objective optimization problem for the maximization of two important bioprocess performance metrics, process yield and productivity. Solving numerically this problem, we find the best available trade-offs between the metrics. Under optimal operation of the mixed community, both strains must produce the protein of interest, and not only one (distribution instead of division of labor). Moreover, in this regime acetate secretion by one strain is necessary for the survival of the other (syntrophy). The results thus illustrate how complex multi-level dynamics shape the optimal production of recombinant proteins by synthetic microbial consortia.