  We explain the relation between certain random tiling models and interacting
particle systems belonging to the anisotropic KPZ (Kardar-Parisi-Zhang)
universality class in 2+1-dimensions. The link between these two \emph{a
priori} disjoint sets of models is a consequence of the presence of shuffling
algorithms that generate random tilings under consideration. To see the precise
connection, we represent both a random tiling and the corresponding particle
system through a set of non-intersecting lines, whose dynamics is induced by
the shuffling algorithm or the particle dynamics. The resulting class of
measures on line ensembles also fits into the framework of the Schur processes.
