  We study the interaction between the inert Higgs doublet (IDM) dark matter
and a vector-like $SU(2)$ triplet lepton (VLL), both of which are $Z_2$-odd.
The vector current of the VLL with the $Z$-boson rules out a fermionic or
two-component dark matter scenario. However, a compressed mass spectrum and a
sufficiently small Yukawa coupling allows co-annihilation and late decay of the
VLL into the IDM sector, affecting the relic density of the pseudoscalar dark
matter. The same two factors enable displaced decay of the VLL states,
providing novel signatures involving hadronically quiet displaced multi-lepton
final states. Such signatures to probe the model are studied at the 14 and 27
TeV LHC, as well as the 100 TeV FCC-hh. In addition to being detectable at the
CMS/ATLAS experiments, if the new particles have sub-100 GeV masses, signals
can also be seen at the proposed MATHUSLA detector.
