Large-scale extragalactic magnetic fields may induce conversions between very-high-energy photons and axionlike particles (ALPs), thereby shielding the photons from absorption on the extragalactic background light. However, in simplified "cell" models, used so far to represent extragalactic magnetic fields, this mechanism would be strongly suppressed by current astrophysical bounds. Here we consider a recent model of extragalactic magnetic fields obtained from large-scale cosmological simulations. Such simulated magnetic fields would have large enhancement in the filaments of matter. As a result, photon-ALP conversions would produce a significant spectral hardening for cosmic TeV photons. This effect would be probed with the upcoming Cherenkov Telescope Array detector. This possible detection would give a unique chance to perform a tomography of the magnetized cosmic web with ALPs.Introduction-Axionlike particles (ALPs) are ultralight pseudoscalar bosons a with a two-photon vertex aγγ, predicted by several extensions of the Standard Model (see [1] for a recent review). In the presence of an external magnetic field, the aγγ coupling leads to the phenomenon of photon-ALP mixing [2]. This effect allows for the possibility of direct searches of ALPs in laboratory experiments. In this respect a rich, diverse experimental program is being carried out, exploiting different sources and approaches [3][4][5]. See [6] for a review.Because the aγγ coupling, ultralight ALPs can also play an important role in astrophysical observations. In particular, an intriguing hint for ALPs has been recently suggested by very-high-energy (VHE) γ-ray experiments. In this respect, recent observations of cosmologically distant γ-ray sources by ground-based γ-ray Imaging Atmospheric Cherenkov Telescopes have revealed a surprising degree of transparency of the Universe to VHE photons [7,8], where one would have expected a significant absorption of VHE photons by pair-production processes (γ VHE + γ EBL → e + e − ) on the extragalactic background light (EBL). Even though this problem has been also analyzed using more conventional physics (see, e.g., [9,10]), photon-ALP oscillations in large-scale magnetic fields provide a natural mechanism to drastically reduce the photon absorption [11][12][13][14][15][16][17][18]. In order to have efficient conversions one should achieve the strong-mixing regime which is realized above a critical energy given by [19]