  Non-minimal Higgs sectors are strongly constrained by the agreement of the
measured couplings of the 125 GeV Higgs with Standard Model predictions. This
agreement can be explained by an approximate $\mathbb{Z}_2$ symmetry under
which the additional Higgs bosons are odd. This allows the additional Higgs
bosons to be approximately inert, meaning that they have suppressed VEVs and
suppressed mixing with the Standard Model Higgs. In this case, single
production of the new Higgs bosons is suppressed, but electroweak pair
production is unsuppressed. We study the phenomenology of a minimal 2 Higgs
doublet model that realizes this scenario. In a wide range of parameters, the
phenomenology of the model is essentially fixed by the masses of the exotic
Higgs bosons, and can therefore be explored systematically. We study a number
of different plausible signals in this model, and show that several LHC
searches can constrain or discover additional Higgs bosons in this parameter
space. We find that the reach is significantly extended at the high luminosity
LHC.
