  Let $f$ and $g$ be commuting meromorphic functions with finitely many poles.
By studying the behaviour of Fatou components under this commuting relation, we
prove that $f$ and $g$ have the same Julia set whenever $f$ and $g$ have no
simply connected fast-escaping wandering domains. By combining this with a
recent result of Tsantaris', we obtain the strongest statement (to date)
regarding the Julia sets of commuting meromorphic functions. In order to
highlight the difference to the entire case, we show that transcendental
meromorphic functions with finitely many poles have orbits that alternate
between approaching a pole and escaping to infinity at strikingly fast rates.
