Although a rigorous theoretical ground on metasurfaces has been established in the recent years on the basis of the equivalence principle, the majority of metasurfaces for converting a propagating wave into a surface wave are developed in accordance with the so-called generalized Snell's law being a simple heuristic rule for performing wave transformations. Recently, Tcvetkova et al. [Phys. Rev. B 97, 115447 (2018)] have rigorously studied this problem by means of a reflecting anisotropic metasurface, which is unfortunately difficult to realize, and no experimental results are available. In this paper, we propose an alternative practical design of a metasurface-based converter by separating the incident plane wave and the surface wave in different half-spaces. It allows one to preserve the polarization of the incident wave and substitute the anisotropic metasurface by an omega-bianisotropic one. The problem is approached from two sides: By directly solving the corresponding boundary problem and by considering the "time-reversed" scenario when a surface wave is converted into a nonuniform plane wave. Particularly, we reveal that an input surface wave plays an important role in the conversion process, influencing the conversion efficiency. To validate the theory, we develop a practical three-layer metasurface based on a conventional printed circuit board technology to mimic the omega-bianisotropic response at the microwave frequency range. The design is verified by full-wave three-dimensional numerical simulations and demonstrates high conversion efficiency. Obtained results are relevant independently of the frequency range and can be generalized to acoustics domain. It enables novel applications, from efficient excitation of waveguide modes in integrated photonic circuits to cloaking of large objects.