The amplitude of redshifted 21 cm fluctuations during the Epoch of Reionization (EoR) is expected to show a distinctive rise and fall behavior with decreasing redshift as reionization proceeds. On large scales (k <~ 0.1 Mpc^{-1}) this can mostly be characterized by evolution in the product of the mean 21 cm brightness temperature and a bias factor, b_21(z). This quantity evolves in a distinctive way that can help in determining the average ionization history of the intergalactic medium (IGM) from upcoming 21 cm fluctuation data sets. Here we consider extracting <T_21> b_21(z) using a combination of future redshifted 21 cm and [CII] line-intensity mapping data sets. Our method exploits the dependence of the 21 cm-[CII]-[CII] cross-bispectrum on the shape of triangle configurations in Fourier space. This allows one to determine <T_21> b_21(z) yet, importantly, is less sensitive to foreground contamination than the 21 cm auto-spectrum, and so can provide a valuable cross-check. We compare the results of simulated bispectra with second-order perturbation theory: on large scales the perturbative estimate of <T_21> b_21(z) matches the true value to within 10% for <x_i> <~ 0.8. We consider the 21 cm auto-bispectrum and show that this statistic may also be used to extract the 21 cm bias factor. Finally, we discuss the survey requirements for measuring the cross-bispectrum. Although we focus on the 21 cm-[CII]-[CII] bispectrum during reionization, our method may be of broader interest and can be applied to any two fields throughout cosmic history.
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