  We investigate the combinatorics of real double Hurwitz numbers with real
positive branch points using the symmetric group. Our main focus is twofold.
First, we prove correspondence theorems relating these numbers to counts of
tropical real covers and study the structure of real double Hurwitz numbers
with the help of the tropical count. Second, we express the numbers as counts
of paths in a subgraph of the Cayley graph of the symmetric group. By
restricting to real double Hurwitz numbers with real positive branch points, we
obtain a concise translation of the counting problem in terms of tuples of
elements of the symmetric group that enables us to uncover the beautiful
combinatorics of these numbers both in tropical geometry and in the Cayley
graph.
