Given a tropical cycle $X$, one can talk about a notion of tropical vector bundles on $X$ having real or tropical fibers. By restricting our attention to bounded rational sections of these bundles, one can develop a good notion of characteristic classes that behave as expected classically. We present further results on these characteristic classes and use it to notably prove the analogue of the splitting principle, which allows us to establish the foundations for Porteous' formula in this setting which provides a determinantal expression for the fundamental class of degeneracy loci in terms of Chern classes.