Tropical geometry produces a combinatorial object called a tropical variety from an algebraic variety. Under certain conditions, geometric invariants can be computed from the tropical variety. We discuss a version of the (pro-unipotent completion of) the fundamental group that can be computed in this manner. This requires introducing new tropical objects that act as vector bundles with flat connections and points and makes use of an extension of the Orlik-Solomon theorem. This is joint work with Kyle Binder.