We consider relativistic, stationary, axisymmetric, polytropic, unconfined, perfect MHD winds, assuming their five Lagrangian first integrals to be known. The asymptotic structure consists of field regions bordered by boundary layers along the polar axis and at null surfaces, such as the equatorial plane, which have the structure of charged column or sheet pinches supported by plasma or magnetic poloidal pressure. In each field-region cell, the proper current (defined here as the ratio of the asymptotic poloidal current to the asymptotic Lorentz factor) remains constant. Our solution is given in the form of matched asymptotic solutions separately valid outside and inside the boundary layers. A Hamilton-Jacobi equation, or equivalently a Grad-Shafranov equation, gives the asymptotic structure in the field regions of winds that carry Poynting flux to infinity. An important consistency relation is found to exist between axial pressure, axial current, and asymptotic Lorentz factor. We similarly derive WKB-type analytic solutions for winds that are kinetic energy-dominated at infinity and whose magnetic surfaces focus to paraboloids. The density on the axis in the polar boundary column is shown to slowly fall off as a negative power of the logarithm of the distance to the wind source. The geometry of magnetic surfaces in all parts of the asymptotic domain, including boundary layers, is explicitly deduced in terms of the first integrals.