  Smooth tropical cubic surfaces are parametrized by maximal cones in the
unimodular secondary fan of the triple tetrahedron. There are $344\, 843 \,867$
such cones, organized into a database of $14\,373\,645$ symmetry classes. The
Schl\"afli fan gives a further refinement of these cones. It reveals all
possible patterns of lines on tropical cubic surfaces, thus serving as a
combinatorial base space for the universal Fano variety. This article develops
the relevant theory and offers a blueprint for the analysis of big data in
tropical geometry.
