  We introduce a new Gentzen-style framework of grafted hypersequents that
combines the formalism of nested sequents with that of hypersequents. To
illustrate the potential of the framework, we present novel calculi for the
modal logics $\mathsf{K5}$ and $\mathsf{KD5}$, as well as for extensions of the
modal logics $\mathsf{K}$ and $\mathsf{KD}$ with the axiom for shift
reflexivity. The latter of these extensions is also known as $\mathsf{SDL}^+$
in the context of deontic logic. All our calculi enjoy syntactic cut
elimination and can be used in backwards proof search procedures of optimal
complexity. The tableaufication of the calculi for $\mathsf{K5}$ and
$\mathsf{KD5}$ yields simplified prefixed tableau calculi for these logic
reminiscent of the simplified tableau system for $\mathsf{S5}$, which might be
of independent interest.
