Two-fluid plasma equations are derived by taking moments of Boltzmann equations. Ignoring collisions and viscous terms and assuming local thermodynamic equilibrium we get five moment equations for each species (electrons and ions), known as two-fluid plasma equations. These equations allow different temperatures and velocities for electrons and ions, unlike ideal magnetohydrodynamics equations. In this article, we present robust second order MUSCL schemes for two-fluid plasma equations based on Strang splitting of the flux and source terms. The source is treated both explicitly and implicitly. These schemes are shown to preserve positivity of the pressure and density. In the case of explicit treatment of source term, we derive explicit condition on the time step for it to be positivity preserving. The implicit treatment of the source term is shown to preserve positivity, unconditionally. Numerical experiments are presented to demonstrate the robustness and efficiency of these schemes.