  This note provides a factorization of a L\'evy pocess over a phase-type
horizon $\tau$ given the phase at the supremum, thereby extending the
Wiener-Hopf factorization for $\tau$ exponential. One of the factors is defined
using time reversal of the phase process. It is shown that there are a variety
of time-reversed representations, all yielding the same factor. Consequences of
this are discussed and examples provided. Additionally, some explicit formulas
for the joint law of the supremum and the terminal value of the process at
$\tau$ are given.
