  We present a Poisson-sigma model describing general 2D dilaton gravity with
non-metricity, torsion and curvature. It involves three arbitrary functions of
the dilaton field, two of which are well-known from metric compatible theories,
while the third one characterizes the local strength of non-metricity. As an
example we show that alpha' corrections in 2D string theory can generate
(target space) non-metricity.
