  The classical statement of Cayley-Salmon that there are 27 lines on every
smooth cubic surface in P^3 fails to hold under tropicalization: a tropical
cubic surface in TP^3 often contains infinitely many tropical lines. Under mild
genericity assumptions, we show that when embedded using the Eckardt triangles
in the anticanonical system, tropical cubic del Pezzo surfaces contain exactly
27 tropical lines. In the non-generic case, which we identify explicitly, we
find up to 27 extra lines, no multiple of which lifts to a curve on the cubic
surface.
  We realize the moduli space of stable anticanonical tropical cubics as a
four-dimensional fan in R^40 with an action of the Weyl group W(E_6). In the
absence of Eckardt points, we show the combinatorial types of these tropical
surfaces are determined by the boundary arrangement of 27 metric trees
corresponding to the tropicalization of the classical 27 lines on the smooth
algebraic cubic surfaces. Tropical convexity and the combinatorics of the root
system E_6 play a central role in our analysis.
