  The classical Poincar\'e formula relates the rational homology classes of
tautological cycles on a Jacobian to powers of the class of Riemann theta
divisor. We prove a tropical analogue of this formula. Along the way, we prove
several foundational results about real tori with integral structures (and,
therefore, tropical abelian varieties). For example, we prove a tropical
version of the Appell-Humbert theorem. We also study various notions of
equivalences between tropical cycles and their relation to one another.
