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We report an experimental and numerical demonstration of dispersive rarefaction shocks (DRS) in a 3D-printed soft chain of hollow elliptical cylinders. We find that, in contrast to conventional nonlinear waves, these DRS have their lower amplitude components travel faster, while the higher amplitude ones propagate slower. This results in the backward-tilted shape of the front of the wave (the rarefaction segment) and the breakage of wave tails into a modulated waveform (the dispersive shock segment). Examining the DRS under various impact conditions, we find the counterintuitive feature that the higher striker velocity causes the slower propagation of the DRS. These unique features can be useful for mitigating impact controllably and efficiently without relying on material damping or plasticity effects.