We construct a multiplicative G-Tate spectral sequence for each R-module X in orthogonal G-spectra, with E²-page given by the Hopf algebra Tate cohomology of R[G]_* with coefficients in π_*(X).
Given a compact Lie group G and a commutative orthogonal ring spectrum R such that R[G]_* = π_*(R ? G₊) is finitely generated and projective over π_*(R), we construct a multiplicative G-Tate spectral sequence for each R-module X in orthogonal G-spectra, with E²-page given by the Hopf algebra Tate cohomology of R[G]_* with coefficients in π_*(X). Under mild hypotheses, such as X being bounded below and the derived page RE∞ vanishing, this spectral sequence converges strongly to the homotopy π_*(XtG) of the G-Tate construction X^tG = [EG ? F(EG+, X]^G.