Let K be an algebraically closed field which is complete with respect to a nontrivial, non-Archimedean valuation and let be its value group. Given a smooth, proper, connected K-curve X and a skeleton of the Berkovich analytification Xan, there are two natural real tori which one can consider: the tropical Jacobian Jac() and the skeleton of the Berkovich analytification Jac(X)an. We show that the skeleton of the Jacobian is canonically isomorphic to the Jacobian of the skeleton as principally polarized tropical abelian varieties. In addition, we show that the tropicalization of a classical Abel-Jacobi map is a tropical Abel-Jacobi map. As a consequence of these results, we deduce that-rational principal divisors on, in the sense of tropical geometry, are exactly the retractions of principal divisors on X. We actually prove a more precise result which says that, although zeros and poles of divisors can cancel under the retraction map, in order to lift a-rational principal divisor on to a principal divisor on X it is never necessary to add more than g extra zeros and g extra poles. Our results imply that a continuous function F: R is the restriction to of.log | f | for some nonzero meromorphic function f on X if and only if F is a-rational tropical meromorphic function, and we use this fact to prove that there is a rational map f: X → P3 whose tropicalization, when restricted to, is an isometry onto its image.