Onset of reconnection in the tail requires the current sheet thickness to be of the order of the ion thermal gyroradius or smaller. However, existing isotropic plasma models cannot explain the formation of such thin sheets at distances where the X-lines are typically observed. Here we reproduce such thin and long sheets in particle-in-cell simulations using a new model of their equilibria with weakly anisotropic ion species assuming quasi-adiabatic ion dynamics, which substantially modifies the current density. It is found that anisotropy/agyrotropy contributions to the force balance in such equilibria are comparable to the pressure gradient in spite of weak ion anisotropy. New equilibria whose current distributions are substantially overstretched compared to the magnetic field lines are found to be stable in spite of the fact that they are substantially longer than isotropic sheets with similar thickness.