  Dark matter could emerge along with the Higgs as a composite
pseudo-Nambu-Goldstone boson $\chi$ with decay constant $f\sim \mathrm{TeV}$.
This type of WIMP is especially compelling because its leading interaction with
the Standard Model, the derivative Higgs portal, has the correct annihilation
strength for thermal freeze-out if $m_\chi \sim O(100)$ GeV, but is negligible
in direct detection experiments due to the very small momentum transfer. The
explicit breaking of the shift symmetry which radiatively generates $m_\chi$,
however, introduces non-derivative DM interactions. In existing realizations a
marginal Higgs portal coupling $\lambda$ is generated with size comparable to
the Higgs quartic, and thus well within reach of XENON1T. Here, we present and
analyze the interesting case where the pattern of explicit symmetry breaking
naturally suppresses $\lambda$ beyond the reach of current and future direct
detection experiments. If the DM acquires mass from bottom quark loops, the
bottom quark also mediates suppressed DM-nucleus scattering with cross sections
that will be eventually probed by LZ. Alternatively, the DM can obtain mass
from gauging its stabilizing $U(1)$ symmetry. No direct detection signal is
expected even at future facilities, but the introduction of a dark photon
$\gamma_D$ has a number of phenomenological implications which we study in
detail, treating $m_{\gamma_D}$ as a free parameter. Complementary probes of
the dark sector include indirect DM detection, DM self-interactions, and extra
radiation, as well as collider experiments. We frame our discussion in an
effective field theory, motivating our parameter choices with a detailed
analysis of an $SO(7)/SO(6)$ composite Higgs model, which can yield either
scenario at low energies.
