  LHC searches with $\tau$ leptons in the final state are always inclusive in
missing-energy sources. A signal in the flavor-violating Higgs decay search,
$h\to\tau\mu$, could therefore equally well be due to a flavor conserving
decay, but with an extended decay topology with additional invisible particles.
We demonstrate this with the three-body decay $h\to\tau\mu\varphi$, where
$\varphi$ is a flavorful mediator decaying to a dark-sector. This scenario can
give thermal relic dark matter that carries lepton flavor charges, a realistic
structure of the charged lepton masses, and explain the anomalous magnetic
moment of the muon, $(g-2)\mu$, while simultaneously obey all indirect
constraints from flavor-changing neutral currents. Another potentially
observable consequence is the broadening of the collinear mass distributions in
the $h\to \tau\mu$ searches.
