What surprised me is that (according to wikipedia at least) the "Tropical" is really just because one of the main dudes inventing it was Brazilian.
It's a really weird topic. This paper seems moderately elementary. This one has some nice pictures. David Corfield links to this one which actually sort of makes me understand why tropical lines look like amoebas: it's because the max-plus algebra actually arises in the limit as you set "planck's constant" to a very small imaginary value, or to put it another way, by doing a little commutator trick with the log/exp functions. Here's a recent bibliography.
