  The K$^{-}$ over K$^{+}$ multiplicity ratio is measured in deep-inelastic
scattering, for the first time for kaons carrying a large fraction $z$ of the
virtual-photon energy. The data were obtained by the COMPASS collaboration
using a 160 GeV muon beam and an isoscalar $^6$LiD target. The regime of
deep-inelastic scattering is ensured by requiring $Q^2>1$ (GeV/$c)^2$ for the
photon virtuality and $W>5$ GeV/$c^2$ for the invariant mass of the produced
hadronic system. Kaons are identified in the momentum range from 12 GeV/$c$ to
40 GeV/$c$, thereby restricting the range in Bjorken-$x$ to $0.01<x<0.40$. The
$z$-dependence of the multiplicity ratio is studied for $z>0.75$. For very
large values of $z$, $i.e.$ $z>0.8$, we observe the kaon multiplicity ratio to
fall below the lower limits expected from calculations based on leading and
next-to-leading order perturbative quantum chromodynamics. Also, the kaon
multiplicity ratio shows a strong dependence on the missing mass of the
single-kaon production process. This suggests that within the perturbative
quantum chromodynamics formalism an additional correction may be required,
which takes into account the phase space available for hadronisation.
