  We explore the concept of real tropical basis of an ideal in the field of
real Puiseux series. We show explicit tropical bases of zero-dimensional real
radical ideals, linear ideals and hypersurfaces coming from combinatorial
patchworking. But we also show that there exist real radical ideals that do not
admit a tropical basis. As an application, we show how to compute the set of
singular points of a real tropical hypersurface. i.e. we compute the real
tropical discriminant.
