Tautological cycles on tropical Jacobians

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