  We establish a new notion of tropical convexity for signed tropical numbers.
We provide several equivalent descriptions involving balance relations and
intersections of open halfspaces as well as the image of a union of polytopes
over Puiseux series and hyperoperations. Along the way, we deduce a new Farkas
lemma and Fourier-Motzkin elimination without the non-negativity restriction on
the variables. This leads to a Minkowski-Weyl theorem for polytopes over the
signed tropical numbers.
