Shocks in gamma-ray emitting classical novae are expected to produce bright thermal and non-thermal X-rays. We test this prediction with simultaneous NuSTAR and Fermi/LAT observations of nova V906 Car, which exhibited the brightest GeV gamma-ray emission to date. The nova is detected in hard X-rays while it is still gamma-ray bright, but contrary to simple theoretical expectations, the detected 3.5-78 keV emission of V906 Car is much weaker than the simultaneously observed >100 MeV emission. No non-thermal X-ray emission is detected, and our deep limits imply that the gamma-rays are likely hadronic. After correcting for substantial absorption (N_H ~ 2 x 10^23 cm^-2), the thermal X-ray luminosity (from a 9 keV optically-thin plasma) is just ~2% of the gamma-ray luminosity. We consider possible explanations for the low thermal X-ray luminosity, including the X-rays being suppressed by corrugated, radiative shock fronts or the X-rays from the gamma-ray producing shock are hidden behind an even larger absorbing column (N_H >10^25 cm^-2). Adding XMM-Newton and Swift/XRT observations to our analysis, we find that the evolution of the intrinsic X-ray absorption requires the nova shell to be expelled 24 days after the outburst onset. The X-ray spectra show that the ejecta are enhanced in nitrogen and oxygen, and the nova occurred on the surface of a CO-type white dwarf. We see no indication of a distinct super-soft phase in the X-ray lightcurve, which, after considering the absorption effects, may point to a low mass of the white dwarf hosting the nova.
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