  We give an explicit graded cellular basis of the $\mathfrak{sl}_3$-web
algebra $K_S$.
  In order to do this, we identify Kuperberg's basis for the
$\mathfrak{sl}_3$-web space $W_S$ with a version of Leclerc-Toffin's
intermediate crystal basis and we identify Brundan, Kleshchev and Wang's degree
of tableaux with the weight of flows on webs and the $q$-degree of foams.
  We use these observations to give a "foamy" version of Hu and Mathas graded
cellular basis of the cyclotomic Hecke algebra which turns out to be a graded
cellular basis of the $\mathfrak{sl}_3$-web algebra.
  We restrict ourselves to the $\mathfrak{sl}_3$ case over $\mathbb{C}$ here,
but our approach should, up to the combinatorics of $\mathfrak{sl}_N$-webs,
work for all $N>1$ or over $\mathbb{Z}$.
