We analyse high-quality NuSTAR observations of the local (z = 0.011) Seyfert 2 active galactic nucleus (AGN) IC 3639, in conjunction with archival Suzaku and Chandra data. This provides the first broadband X-ray spectral analysis of the source, spanning nearly two decades in energy (0.5-30 keV). Previous X-ray observations of the source below 10 keV indicated strong reflection/obscuration on the basis of a pronounced iron fluorescence line at 6.4 keV. The hard X-ray energy coverage of NuSTAR, together with self-consistent toroidal reprocessing models, enables direct broadband constraints on the obscuring column density of the source. We find the source to be heavily Compton-thick (CTK) with an obscuring column in excess of $3.6times10^{24}$ cm$^{-2}$, unconstrained at the upper end. We further find an intrinsic 2-10 keV luminosity of $textrm{log}_{10}(L_{textrm{2-10 keV}} textrm{[erg s}^{-1}]) = 43.4^{+0.6}_{-1.1}$ to 90% confidence, almost 400 times the observed flux, and consistent with various multi-wavelength diagnostics. Such a high intrinsic to observed flux ratio in addition to an Fe-K$alpha$ fluorescence line equivalent width exceeding 2 keV is extreme amongst known bona fide CTK AGN, which we suggest are both due to the high level of obscuration present around IC 3639. Our study demonstrates that broadband spectroscopic modelling with NuSTAR enables large corrections for obscuration to be carried out robustly, and emphasises the need for improved modelling of AGN tori showing intense iron fluorescence.
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