We present an analysis of the anisotropic redshift-space void-galaxy correlation in configuration space using the Sloan Digital Sky Survey extended Baryon Oscillation Spectroscopic Survey (eBOSS) Data Release 16 luminous red galaxy (LRG) sample. This sample consists of LRGs between redshifts 0.6 and 1.0, combined with the high redshift $z>0.6$ tail of the Baryon Oscillation Spectroscopic Survey Data Release 12 CMASS sample. We use a reconstruction method to undo redshift-space distortion (RSD) effects from the galaxy field before applying a watershed void-finding algorithm to remove bias from the void selection. We then perform a joint fit to the multipole moments of the correlation function for the growth rate $fsigma_8$ and the geometrical distance ratio $D_M/D_H$, finding $fsigma_8(z_mathrm{eff})=0.356pm0.079$ and $D_M/D_H(z_mathrm{eff})=0.868pm0.017$ at the effective redshift $z_mathrm{eff}=0.69$ of the sample. The posterior parameter degeneracies are orthogonal to those from galaxy clustering analyses applied to the same data, and the constraint achieved on $D_M/D_H$ is significantly tighter. In combination with the consensus galaxy BAO and full-shape analyses of the same sample, we obtain $fsigma_8=0.447pm0.039$, $D_M/r_d=17.48pm0.23$ and $D_H/r_d=20.10pm0.34$. These values are in good agreement with the $Lambda$CDM model predictions and represent reductions in the uncertainties of $13%$, $23%$ and $28%$ respectively compared to the combined results from galaxy clustering, or an overall reduction of 55% in the allowed volume of parameter space.
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