  Kreweras words are words consisting of n A's, n B's, and n C's in which every
prefix has at least as many A's as B's and at least as many A's as C's.
Equivalently, a Kreweras word is a linear extension of the poset ${\sf V}\times
[n]$. Kreweras words were introduced in 1965 by Kreweras, who gave a remarkable
product formula for their enumeration. Subsequently they became a fundamental
example in the theory of lattice walks in the quarter plane. We study
Sch\"{u}tzenberger's promotion operator on the set of Kreweras words. In
particular, we show that 3n applications of promotion on a Kreweras word merely
swaps the B's and C's. Doing so, we provide the first answer to a question of
Stanley from 2009, asking for posets with `good' behavior under promotion,
other than the four families of shapes classified by Haiman in 1992. We also
uncover a strikingly simple description of Kreweras words in terms of
Kuperberg's $\mathfrak{sl}_3$-webs, and Postnikov's trip permutation associated
with any plabic graph. In this description, Sch\"{u}tzenberger's promotion
corresponds to rotation of the web.
