Tangential current sheets are ubiquitous in space plasmas and yet hard to describe with a kinetic equilibrium. In this paper, we use a semi-analytical model, the BAS model, which provides a steady ion distribution function for a tangential asymmetric current sheet and we prove that an ion kinetic equilibrium produced by this model remains steady in a fully kinetic particle-in-cell simulation even if the electron distribution function does not satisfy the time independent Vlasov equation. We then apply this equilibrium to look at the dependence of magnetic reconnection simulations on their initial conditions. We show that, as the current sheet evolves from a symmetric to an asymmetric upstream plasma, the reconnection rate is impacted and the X line and the electron flow stagnation point separate from one another and start to drift. For the simulated systems, we investigate the overall evolution of the reconnection process via the classical signatures discussed in the literature and searched in the Magnetospheric MultiScale data. We show that they seem robust and do not depend on the specific details of the internal structure of the initial current sheet.