Different components of real sections for twistor spaces of hyper-Kähler manifolds

Question

A hyper-Kähler manifold is a Riemannian manifold $(M,g)$ equipped with 3 complex structures $I,J,K$ obeying the quaternionic relations and such that $g$ is a Kähler metric for each complex structure. In ''Hyperkähler metrics and super-symmetry'' (Comm. Math. Phys. 108 (1987)), Hitchin, Karlhede, Lindström and Rocek have shown that the hyperkähler structure is encoded in terms of complex analytic data of the associtaed twistor fibration $\mathcal P\to\mathbb CP^1$, where $\mathcal P=M\times S^2$ is equipped with a natural complex structure (induced by the hyperkähler structure and parametrizing the 2-sphere of complex structures). In particular, if $g$ is complete then $M$ is a component of real holomorphic sections of $\mathcal P\to\mathbb CP^1$, where the real structure is given by $$(p,x)\in M\times S^2=\mathcal P\mapsto (p,-x)\in\mathcal P.$$ I would like to know of examples (and/or references to them) of twistor spaces of hyperkähler manifolds which have different components of real holomorphic sections. Some of my recent work is about the construction of such examples, and I would like to know about others. I am sure there must have been some work in this direction, but I was not able to find anything.