  B\'ar\'any's colorful generalization of Carath\'eodory's Theorem combines
geometrical and combinatorial constraints. Kalai-Meshulam (2005) and Holmsen
(2016) generalized B\'ar\'any's theorem by replacing color classes with matroid
constraints. In this note, we obtain corresponding results in tropical
convexity, generalizing the tropical colorful Carath\'eodory Theorem of
Gaubert-Meunier (2010). Our proof is inspired by geometric arguments and is
reminiscent of matroid intersection. In particular, we show that the
topological approach fails in this setting. We also discuss tropical colorful
linear programming and show that it is NP-complete. We end with thoughts and
questions on generalizations to polymatroids, anti-matroids as well as examples
and matroid simplicial depth.
