  It was recently discovered that Killing horizons in the generic
Kerr-NUT-(anti) de Sitter spacetimes are projectively singular, i.e. their
spaces of the null generators have singular geometry. Only if the cosmological
constant takes the special value determined by the Kerr and NUT parameters, and
the radius of the horizon, then the corresponding horizon does not suffer that
problem. In the current paper, the projectively non-singular horizons are
investigated. They are found to be cosmological and non-extremal. Every
projectively non-singular horizon can be used to define a global completion of
the Kerr-NUT-de Sitter spacetime it is contained in. The resulting spacetime
extends from $\mathcal{I}^-$ to $\mathcal{I}^+$, has the topology of
$\mathbb{R}\times S_3$ and is smooth except for a possible Kerr-like
singularity.
