Context. The isolated neutron star (INS) 2XMM J104608.7-594306 is one of the only two to be discovered through their thermal emission since the ROSAT era. Possibly a remnant of a former generation of massive stars in the Carina nebula, the exact nature of the source is unclear, and it might be unique amongst the several classes of Galactic INSs. Aims. In a first dedicated XMM-Newton observation of the source, we found intriguing evidence of a very fast spin period of P ∼ 18.6 ms at the 4σ confidence level. Moreover, spectral features in absorption have also been identified. We re-observed 2XMM J104608.7-594306 with XMM-Newton to better characterise the spectral energy distribution of the source, confirm the candidate spin period, and possibly constrain the pulsar spin-down. Methods. We used the two XMM-Newton observations of 2XMM J104608.7-594306 to perform detailed timing and spectral X-ray analysis. Both the spin-down rate and the energy of the spectral features provide estimates on the neutron star magnetic field, which are crucial for investigating the evolutionary state of the neutron star. Results. Statistically acceptable spectral fits and meaningful physical parameters for the source are only obtained when the residuals at energies 0.55 keV and 1.35 keV are taken into account by the spectral modelling. While the former can result from the inhomogeneous temperature distribution on the surface of the neutron star or can be related to a local overabundance of oxygen in the Carina nebula, the one at 1.35 keV is only satisfactorily accounted for by invoking a line in absorption. In this case, the best-fit neutron star atmosphere models constrain the hydrogen column density, the effective temperature, and the luminosity of the source within N H = (2.5−3.3) × 10 21 cm −2 , T eff = (6−10) × 10 5 K, and L X = (1.1−7.4) × 10 32 erg s −1 . The implied distance is consistent with a location in (or in front of) the Carina nebula, and radiation radii are compatible with emission originating on most of the surface. Non-thermal X-ray emission is ruled out at levels above 0.5% (3σ) of the source luminosity. Unfortunately, the second XMM-Newton observation proved inconclusive in terms of confirming (discarding) the fast candidate spin, providing an upper limit on the pulsed fraction of the source that is very close to the limiting sensitivity for detecting the modulation found previously. Conclusions. In the absence of an unambiguous period determination and an estimate of the magnetic field, the nature of the source remains open to interpretation. Its likely association with the Carina cluster and its overall spectral properties (only shared by a handful of other peculiar INSs) disfavour a standard evolutionary path or one in which the source was previously recycled by accretion in a binary system. The star 2XMM J104608.7-594306 may be similar to Calvera (1RXS J141256.0+792204), a neutron star for which the scenario of an evolved anti-magnetar has been discussed. A better age estimate and deeper radio and γ-ray l...