Joint probability distribution ℬ(x, δ_dm) of halo properties x (number counts, virial mass and velocity dispersion) and the underlying dark matter density, interpolated on a 192³ mesh using a CIC mass-assignment scheme and for different cosmic-web environments. Solid lines (and coloured regions) denote contours enclosing 98% and 68% of the total number of cells in a reference SLICS simulation. Dotted lines represent the same quantity obtained from one BAM halo catalogue.