  We consider scalar extensions of the SM and their effective field theoretic
generalisations to illustrate the phenomenological connection between precision
measurements of the anomalous magnetic moment of the muon $a_\mu$, precision
Higgs measurements, and direct collider sensitivity. To this end, we consider
charged BSM scalar sectors of the Zee-Babu type for which we develop a
consistent, and complete dimensions-5 and -6 effective field theory extension.
This enables us to track generic new physics effects that interact with the SM
predominantly via radiative interactions. While the operator space is high
dimensional, the intersection of exotics searches at the Large Hadron Collider,
Higgs signal strength and anomalous muon magnetic measurements is manageably
small. We find that consistency of LHC Higgs observations and $a_\mu$ requires
a significant deformation of the new states' electroweak properties. Evidence
in searches for doubly charged scalars as currently pursued by the LHC
experiments can be used to further tension the BSMEFT parameter space and
resolve blind directions in the EFT-extended Zee-Babu scenario.
