  The twistor space of a Riemannian 4-manifold carries two almost complex
structures, $J_+$ and $J_-$, and a natural closed 2-form $\omega$. This article
studies limits of manifolds for which $\omega$ tames either $J_+$ or $J_-$.
This amounts to a curvature inequality involving self-dual Weyl curvature and
Ricci curvature, and which is satisfied, for example, by all anti-self-dual
Einstein manifolds with non-zero scalar curvature. We prove that if a sequence
of manifolds satisfying the curvature inequality converges to a hyperk\"ahler
limit X (in the $C^2$ pointed topology) then X cannot contain a holomorphic
2-sphere (for any of its hyperk\"ahler complex structures). In particular, this
rules out the formation of bubbles modelled on ALE gravitational instantons in
such families of metrics.
