A cylindrical synthetic antiferromagnet is proposed as a potential candidate to generate nonreciprocal spin waves. It is demonstrated that such a system presents a notable spin-wave asymmetry induced by the combined action of the antiferromagnetic state and the curvature. Analytical expressions are proposed for the case of thin cylindrical shells. These results are relevant from a fundamental and practical point of view since the chirality of the spin waves is a crucial ingredient to visualize future magnon-based logic applications.