For the first time, the $K^-/K^+$ multiplicity ratio is measured in deep-inelastic scattering 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. The Bjorken scaling variable range is $0.01<x<0.40$. The $z$-dependence of the multiplicity ratio is studied for $z>0.75$. For very large values of $z$, {\it i.e.} $z>0.8$, the results contradict expectations obtained using the formalism of (next-to-)leading order perturbative quantum chromodynamics. Our studies suggest that, within this formalism, an additional correction may be required to take into account the phase space available for hadronisation.