Magnetic thin films with Dzyaloshinskii-Moriya interactions (DMI) are receiving enormous interest because of recent developments in the understanding of DMI's role in controlling the efficiency of domain wall (DW) motion  and in creating magnetic skyrmions . Recently, interfacial DMI has been proposed to be directly related to Rashba spin-orbit coupling at magnetic interfaces . Based on this theory, magnetic domain-wall (DW) resistance is predicted to gain an additional term originating from the Rashba effect. We call this extra term 'chiral resistance', as the sign opposite for DWs with different chiralities, which means that electrons have different scattering rates when moving through DWs with positive and negative chiralities. Since this chiral resistance solely originates from Rashba effect, a direct observation can help to illuminate the debated roles of DMI and Rashba versus Spin-Hall effects for fast current-driven domain wall motion . Moreover, this chiral resistance can be very useful for designing future energy-efficient domain wall devices, e.g. if we combine it with the recently proposed electric-field control of DW chirality .