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The stability of the MHD tangential discontinuity is studied in compressible plasmas in the presence of anisotropic viscosity and thermal conductivity. The general dispersion equation is derived and solutions to this dispersion equation and stability criteria are obtained for the limiting cases of incompressible and cold plasmas. In these two limiting cases the effect of thermal conductivity vanishes and the solutions are only influenced by viscosity. The stability criteria for viscous plasmas are compared with those for ideal plasmas where stability is determined by the Kelvin-Helmholtz velocity V_{KH} as a threshold for the difference in the equilibrium velocities. Viscosity turns out to have a destabilizing influence when the viscosity coefficient takes different values at the two sides of the discontinuity. Viscosity lowers the threshold velocity V_c below the ideal Kelvin-Helmholtz velocity V_{KH}, so that there is a range of velocities between V_c and V_{KH} where the overstability is of a dissipative nature.