  We study the monoid generated by n-by-n distance matrices under tropical (or
min-plus) multiplication. Using the tropical geometry of the orthogonal group,
we prove that this monoid is a finite polyhedral fan of dimension n(n-1)/2, and
we compute the structure of this fan for n up to 5. The monoid captures gossip
among n gossipers over lossy phone lines, and contains the gossip monoid over
ordinary phone lines as a submonoid. We prove several new results about this
submonoid, as well. In particular, we establish a sharp bound on chains of
calls in each of which someone learns something new.
