This work investigates the connection between the cosmic web and the halo distribution through the gravitational potential at the field level. We combine three fields of research, cosmic web classification, perturbation theory expansions of the halo bias, and halo (galaxy) mock catalogue making methods. In particular, we use the invariants of the tidal field and the velocity shear tensor as generating functions to reproduce the halo number counts of a reference catalogue from full gravity calculations, populating the dark matter field on a mesh well into the non-linear regime ($3\, h^{-1}\, {\rm Mpc}$ scales). Our results show an unprecedented agreement with the reference power spectrum within 1 per cent up to $k=0.72\, h\, {\rm Mpc}^{-1}$. By analysing the three-point statistics on large scales (configurations of up to $k=0.2\, h\, {\rm Mpc}^{-1}$), we find evidence for non-local bias at the 4.8σ confidence level, being compatible with the reference catalogue. In particular, we find that a detailed description of tidal anisotropic clustering on large scales is crucial to achieve this accuracy at the field level. These findings can be particularly important for the analysis of the next generation of galaxy surveys in mock galaxy production.