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We list up all the candidates for the real isotopy types of realanti-bicanonical curves with one real nondegenerate double point on the 4-threal Hirzebruch surface RF_4 by enumerating the connected components of themoduli space of real 2-elementary K3 surfaces of type (S,\theta)=((3,1,1),-id). We also list up all the candidates for the non-increasing simplestdegenerations of real nonsingular anti-bicanonical curves on RF_4. We find aninteresting correspondence between the real isotopy types of realanti-bicanonical curves with one real nondegenerate double point on RF_4 andthe non-increasing simplest degenerations of real nonsingular anti-bicanonicalcurves on RF_4. This correspondence is very similar to the one provided by therigid isotopic classification of real sextic curves on RP^2 with one realnondegenerate double point by I. Itenberg.

On real anti-bicanonical curves with one double point on the 4-th real Hirzebruch surface