  Tropical polyhedra are known to be representable externally, as intersections
of finitely many tropical half-spaces. However, unlike in the classical case,
the extreme rays of their polar cones provide external representations
containing in general superfluous half-spaces. In this paper, we prove that any
tropical polyhedral cone in R^n (also known as "tropical polytope" in the
literature) admits an essentially unique minimal external representation. The
result is obtained by establishing a (partial) anti-exchange property of
half-spaces. Moreover, we show that the apices of the half-spaces appearing in
such non-redundant external representations are vertices of the cell complex
associated with the polyhedral cone. We also establish a necessary condition
for a vertex of this cell complex to be the apex of a non-redundant half-space.
It is shown that this condition is sufficient for a dense class of polyhedral
cones having "generic extremities".
