  The increasing tritronqu\'ee solutions of the Painlev\'e-II equation with
parameter $\alpha$ exhibit square-root asymptotics in the maximally-large
sector $|\arg(x)|<\tfrac{2}{3}\pi$ and have recently appeared in applications
where it is necessary to understand the behavior of these solutions for complex
values of $\alpha$. Here these solutions are investigated from the point of
view of a Riemann-Hilbert representation related to the Lax pair of Jimbo and
Miwa, which naturally arises in the analysis of rogue waves of infinite order.
We show that for generic complex $\alpha$, all such solutions are
asymptotically pole-free along the bisecting ray of the complementary sector
$|\arg(-x)|<\tfrac{1}{3}\pi$ that contains the poles far from the origin. This
allows the definition of a total integral of the solution along the axis
containing the bisecting ray, in which certain algebraic terms are subtracted
at infinity and the poles are dealt with in the principal-value sense. We
compute the value of this integral for all such solutions. We also prove that
if the Painlev\'e-II parameter $\alpha$ is of the form
$\alpha=\pm\tfrac{1}{2}+{\rm i}p$, $p\in\mathbb{R}\setminus\{0\}$, one of the
increasing tritronqu\'ee solutions has no poles or zeros whatsoever along the
bisecting axis.
