  We study the collider phenomenology of a neutral gauge boson $Z'$ arising in
minimal but anomalous $\mathrm{U}(1)$ extensions of the Standard Model (SM). To
retain gauge invariance of physical observables, we consider cancellation of
gauge anomalies through the Green-Schwarz mechanism. We categorize a wide class
of $\mathrm{U}(1)$ extensions in terms of the new $\mathrm{U}(1)$ charges of
the left-handed quarks and leptons and the Higgs doublet. We derive constraints
on some benchmark models using electroweak precision constraints and the latest
13 TeV LHC dilepton and dijet resonance search data. We calculate the decay
rates of the exotic and rare one-loop $Z'$ decays to $ZZ$ and $Z$-photon modes,
which are the unique signatures of our framework. If observed, these decays
could hint at anomaly cancellation through the Green-Schwarz mechanism. We also
discuss the possible observation of such signatures at the LHC and at future
ILC colliders.
