In advective transport through weakly heterogeneous aquifers of random stationary and isotropic three-dimensional permeability distribution, transverse macrodispersivity αT is found to be zero. This was determined in the past by solving the transport equation at first order in the log conductivity variance σ2γ. However, field findings indicate the presence of small but finite αT . The aim of the paper is to determine αT for highly heterogeneous formations using a model that contains inclusions of conductivity K, submerged in a matrix of conductivity K0, for large κ = K/K0. In the dilute medium approximation, valid for small volume fraction n, but arbitrary κ, and for spherical inclusions, it is found that αT = 0 because of the axisymmetry of flow past a sphere. A medium made up of rotational ellipsoids of arbitrary random orientation, macroscopically isotropic, and of the same κ and n is devised as a counterexample. It is found that because of the intertwining of streamlines αT > 0, being of order (κ - 1)4 for κ » 1. These findings are confirmed by accurate numerical simulations of flow through a large number of interacting inclusions; for κ = 10 and n = 0.2 (jamming limit), the large value αT/α T≃ 0-15 is attained. The numerical simulations display the strong permanent deformation of stream tubes responsible for this phenomenon, corned as "advective mixing." The two-point covariance, used in practice in order to characterize the aquifer structure, is not able to detect the structures that produce advective mixing. Nevertheless, the presence of high-conductivity lenses inclined with respect to the mean flow may explain the occurrence of finite αT in field applications.