The first measurements of the coherence factor R_{K_S^0Kpi} and the average strong--phase difference delta^{K_S^0Kpi} in D^0 to K_S^0 K^mppi^pm decays are reported. These parameters can be used to improve the determination of the unitary triangle angle gamma in B^- rightarrow $widetilde{D}K^-$ decays, where $widetilde{D}$ is either a D^0 or a D^0-bar meson decaying to the same final state, and also in studies of charm mixing. The measurements of the coherence factor and strong-phase difference are made using quantum-correlated, fully-reconstructed D^0D^0-bar pairs produced in e^+e^- collisions at the psi(3770) resonance. The measured values are R_{K_S^0Kpi} = 0.70 pm 0.08 and delta^{K_S^0Kpi} = (0.1 pm 15.7)$^circ$ for an unrestricted kinematic region and R_{K*K} = 0.94 pm 0.12 and delta^{K*K} = (-16.6 pm 18.4)$^circ$ for a region where the combined K_S^0 pi^pm invariant mass is within 100 MeV/c^2 of the K^{*}(892)^pm mass. These results indicate a significant level of coherence in the decay. In addition, isobar models are presented for the two decays, which show the dominance of the K^*(892)^pm resonance. The branching ratio {B}(D^0 rightarrow K_S^0K^+pi^-)/{B}(D^0 rightarrow K_S^0K^-pi^+) is determined to be 0.592 pm 0.044 (stat.) pm 0.018 (syst.), which is more precise than previous measurements.
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