Here we start with two complete noncompact hyperkähler $4$-manifolds with cylindrical ends. These were constructed by Tian and Yau [TY90] by removing smooth fibers from rational elliptic surfaces, and were proved in [Hei12] to converge to their $\mathbb{R}\times\mathbb{T}^{3}$ flat asymptotic models at an exponential rate. Such spaces are known as $\ALH$-spaces or half-$\K 3$ surfaces in the literature. It is then possible to glue together two $\ALH$ spaces to obtain a family of hyperkähler metrics on $\K 3$ which degenerates by developing a long neck modeled on $\mathbb{T}^{3}$ times an interval (see [CC16] for a rigorous proof). If we rescale these metrics so that the rescaled diameter equals $1$, then the Gromov-Hausdorff limit is the unit interval and the bubbles are the Tian-Yau asymptotically cylindrical metrics at each endpoint. Gluing of asymptotically cylindrical geometric structures is a very familiar construction in geometry, see for example [Flo91, KS01] for anti-self-dual metrics in dimension $4$, and [Kov03] for holonomy $G_{2}$ metrics in dimension $7$.