We report measurements of the flux-integrated |$\overline{\nu}_\mu$| and |$\overline{\nu}_\mu+\nu_\mu$| charged-current cross-sections on water and hydrocarbon targets using the T2K anti-neutrino beam with a mean beam energy of 0.86 GeV. The signal is defined as the (anti-)neutrino charged-current interaction with one induced |$\mu^\pm$| and no detected charged pion or proton. These measurements are performed using a new WAGASCI module recently added to the T2K setup in combination with the INGRID Proton Module. The phase space of muons is restricted to the high-detection efficiency region, |$p_{\mu}>400~{\rm MeV}/c$| and |$\theta_{\mu}<30^{\circ}$|⁠, in the laboratory frame. An absence of pions and protons in the detectable phase spaces of |$p_{\pi}>200~{\rm MeV}/c$|⁠, |$\theta_{\pi}<70^{\circ}$| and |$p_{\rm p}>600~{\rm MeV}/c$|⁠, |$\theta_{\rm p}<70^{\circ}$| is required. In this paper, both the |$\overline{\nu}_\mu$| cross-sections and |$\overline{\nu}_\mu+\nu_\mu$| cross-sections on water and hydrocarbon targets and their ratios are provided by using the D’Agostini unfolding method. The results of the integrated |$\overline{\nu}_\mu$| cross-section measurements over this phase space are |$\sigma_{\rm H_{2}O}=(1.082\pm0.068(\rm stat.)^{+0.145}_{-0.128}(\rm syst.)) \times 10^{-39}\,{\rm cm^{2} / nucleon}$|⁠, |$\sigma_{\rm CH}=(1.096\pm0.054(\rm stat.)^{+0.132}_{-0.117}(\rm syst.)) \times 10^{-39}\,{\rm cm^{2} / nucleon}$|⁠, and |$\sigma_{\rm H_{2}O}/\sigma_{\rm CH} = 0.987\pm0.078(\rm stat.)^{+0.093}_{-0.090}(\rm syst.)$|⁠. The |$\overline{\nu}_\mu+\nu_\mu$| cross-section is |$\sigma_{\rm H_{2}O} = (1.155\pm0.064(\rm stat.)^{+0.148}_{-0.129}(\rm syst.)) \times 10^{-39}\,{\rm cm^{2} / nucleon}$|⁠, |$\sigma_{\rm CH}=(1.159\pm0.049(\rm stat.)^{+0.129}_{-0.115}(\rm syst.)) \times 10^{-39}\,{\rm cm^{2} / nucleon}$|⁠, and |$\sigma_{\rm H_{2}O}/\sigma_{\rm CH}=0.996\pm0.069(\rm stat.)^{+0.083}_{-0.078}(\rm syst.)$|⁠.