Abstract: We report measurements of the flux-integrated (nu) over bar (mu) and (nu) over bar (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(+/-) 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 MeV/c and theta(mu) < 30 degrees, in the laboratory frame. An absence of pions and protons in the detectable phase spaces of p(pi) > 200 MeV/c, theta(pi) < 70 degrees and p(p) > 600 MeV/c, theta(p) < 70 degrees is required. In this paper, both the <(nu)over bar>(mu), cross-sections and (nu) over bar (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 (nu) over bar (mu), cross-section measurements over this phase space are sigma(H2O) = (1.082 +/- 0.068(stat.)(+0.145)(-0.128)(syst.)) x 10(-39) cm(2)/nucleon, sigma(CH) = (1.096 +/- 0.054 (stat.)(+0.132)(-0.117)(syst.)) x 10(-39) cm(2) /nucleon, and sigma(H2O)/sigma(CH) = 0.987 +/- 0.078 (stat.)(+0.093)(-0.090)(syst.). The (nu) over bar (mu), + nu(mu), cross-section is sigma(H2O) = (1.155 +/- 0.064(stat.)(+0.148)(-0.129)(syst.)) x 10(-39) cm(2)/nucleon, sigma(CH) = (1.159 +/- 0.049(stat.)(+0.129)(-0.115)(syst.)) x 10(-39) cm(2)/nucleon, and sigma(H2O)/sigma(CH) = 0.996 +/- 0.069(stat.)(+0.083)(-0.078)(syst.).