Cosmic web anisotropy is the primary indicator of halo assembly bias
  The internal properties of dark matter haloes correlate with the large-scale
halo clustering strength at fixed halo mass $-$ an effect known as assembly
bias $-$ and are also strongly affected by the local, non-linear cosmic web.
Characterising a halo's local web environment by its tidal anisotropy $\alpha$
at scales $\sim4$ x the halo radius, we demonstrate that these multi-scale
correlations represent two distinct statistical links: one between the internal
property and $\alpha$, and the other between $\alpha$ and large-scale (
$>30h^{-1}$Mpc) halo bias $b_1$. We focus on scalar internal properties of
haloes related to formation time (concentration $c_{\rm vir}$), shape (mass
ellipsoid asphericity $c/a$), velocity dispersion structure (velocity ellipsoid
asphericity $c_v/a_v$ and velocity anisotropy $\beta$) and angular momentum
(dimensionless spin $\lambda$) in the mass range $8\times10^{11}< M_{\rm
vir}/(h^{-1}M_\odot)<5\times10^{14}$. Using conditional correlation
coefficients and other detailed tests, we show that the joint distribution of
$\alpha$, $b_1$ and any of the internal properties
$c\in\{\beta,c_v/a_v,c/a,c_{\rm vir},\lambda\}$ is consistent with
$p(\alpha,b_1,c)\simeq p(\alpha)p(b_1|\alpha)p(c|\alpha)$, at all but the
largest masses. $\textit{Thus, the assembly bias trends $c-b_1$ reflect the two
fundamental correlations $c-\alpha$ and $b_1-\alpha$.}$ Our results are
unaffected by the exclusion of haloes with recent major merger events or
splashback objects, although the latter are distinguished by the fact that
$\alpha$ does not explain their assembly bias trends. The overarching
importance of $\alpha$ provides a new perspective on the nature of assembly
bias of distinct haloes, with potential ramifications for incorporating
realistic assembly bias effects into mock catalogs of future large-scale
structure surveys and for detecting galaxy assembly bias.
