Spin–orbit torque that originates from spin Hall effect and Dzyaloshinskii–Moriya interaction (DMI) can efficiently move chiral magnetic domain walls in perpendicularly magnetized wires. It has been shown that antiferromagnetically coupled composite domain walls across a ruthenium layer can be driven even faster by exchange coupling torque that is proportional to exchange coupling strength. Here, we report a current-driven motion of composite chiral domain walls in synthetic antiferromagnets with a rhodium spacer layer. It is found that the domain walls in the wire with a rhodium layer do not move as fast as that with a ruthenium layer although the exchange coupling in Co|Rh|Co is stronger than Co|Ru|Co, which is due to the formation of a large DMI at the Rh|Co interface. The Dzyaloshinskii–Moriya interaction at the Co/Rh interface has the same sign and comparable strength to the Pt|Co interface, thus negating the exchange coupling torque. The spin Hall effect from rhodium is found to be as small as ruthenium. Our findings show that rhodium can be used to tailor the DMI strengths in the current-driven motion of chiral domain walls in various magnetic nanostructures.