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A real 2-elementary K3 surfaces of type ((3,1,1),- id) yields a realanti-bicanonical curve s \cup A^\prime_1 (disjoint union) on the 4-th realHirzebruch surface F_4 where s is the exceptional section of F_4 and the realcurve A^\prime_1 has one real double point. We give a criterion (seeProposition 2.4) which determines whether the real double point is degenerateor not. One direction of the assertion of this proposition has already beenproved in Lemma 4.6 of the preceding paper [9] (2015) by the author. In thispaper we prove the inverse direction.

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