Neutron diffraction and magnetization studies of short-period antiferromagnetic (AF) semiconducting (111) EuTe/PbTe superlattices are presented. Due to the symmetry-breaking lattice mismatch strain, only a single AF configuration forms in the EuTe layers, instead of four allowed by symmetry in bulk EuTe. The strain and the finite layer thickness also lead to a strong shift in the Neel temperature. Neutron-diffraction spectra exhibit pronounced patterns of satellites, indicating spin correlations between successive EuTe layers separated by PbTe layers, and the transfer of magnetic interactions across the diamagnetic spaces. Experiments on samples with doped PbTe layers and in external magnetic fields rule out that the coupling is caused by any of the mechanisms known to occur in metallic superlattices. Hence, our data strongly suggest the existence of a new interlayer coupling mechanism.