  We reprove the strong Hanani-Tutte theorem on the projective plane. In
contrast to the previous proof by Pelsmajer, Schaefer and Stasi, our method is
constructive and does not rely on the characterization of forbidden minors,
which gives hope to extend it to other surfaces. Moreover, our approach can be
used to provide an efficient algorithm turning a Hanani-Tutte drawing on the
projective plane into an embedding.
