  We consider the quasi-Ablowitz-Segur and quasi-Hastings-McLeod solutions of
the inhomogeneous Painlev\'e II equation $$ u"(x)=2u^3(x)+xu(x)-\alpha \qquad
\textrm{for } \alpha \in \mathbb{R} \textrm{ and } |\alpha| > \frac{1}{2}. $$
These solutions are obtained from the classical Ablowitz-Segur and
Hastings-McLeod solutions via the B\"acklund transformation, and satisfy the
same asymptotic behaviors when $x \to \pm \infty$. For $|\alpha| > 1/2$, we
show that the quasi-Ablowitz-Segur and quasi-Hastings-McLeod solutions possess
$[ \, |\alpha| + \frac{1}{2} \, ]$ simple poles on the real axis, which
rigorously justifies the numerical results in Fornberg and Weideman (Found.
Comput. Math., 14 (2014), no. 5, 985-1016).
