A spinor satisfying a reality condition is a Majorana spinor and the question is about the existence of a Majorana-Weyl spinor. But in four spacetime dimensions (in Lorentzian signature), no Majorana-Weyl spinor exists, the Weyl spinor is the smallest and so the minimal supersymmetry in four dimensions has four real supercharges and the scalar superpartner of a Weyl spinor has to be a complex scalar.

It is possible to find something analogous in three spacetime dimensions. In this case, the smallest spinor is a Majorana spinor, with two real components, and so the minimal supersymmetry in three dimensions has two real supercharges (half the number of the minimal supersymmetry in four dimensions) and the scalar superpartner of a Majorana spinor in three dimensions is a real scalar.