Tropical Poincar\'e bundle, Fourier-Mukai transform, and a generalized Poincar\'e formula

We construct a tropical analogue of the Poincar\'e bundle and prove a (cohomological) Fourier-Mukai transform for real tori with integral structures. We then prove a tropical analogue of Beauville's generalized Poincar\'e formula for polarized abelian varieties. Some consequences include a geometric Riemann-Roch theorem for tropical abelian varieties, as well as a tropical Poincar\'e-Prym formula which was recently conjectured by R\"ohrle and Zakharov.