  We use tropical and nonarchimedean geometry to study the moduli space of
genus $0$ stable maps to $\mathbb{P}^1$ relative to two points. This space is
exhibited as a tropical compactification in a toric variety. Moreover, the fan
of this toric variety may be interpreted as a moduli space for tropical
relative stable maps with the same discrete data. As a consequence, we confirm
an expectation of Bertram and the first two authors, that the tropical Hurwitz
cycles are tropicalizations of classical Hurwitz cycles. As a second
application, we obtain a full descendant correspondence for genus $0$ relative
invariants of $\mathbb{P}^1$.
