  We consider a Markov additive process with a finite phase space and study its
path decompositions at the times of extrema, first passage and last exit. For
these three families of times we establish splitting conditional on the phase,
and provide various relations between the laws of post- and pre-splitting
processes using time reversal. These results offer valuable insight into
behavior of the process, and while being structurally similar to the L\'evy
process case, they demonstrate various new features. As an application we
formulate the Wiener-Hopf factorization, where time is counted in each phase
separately and killing of the process is phase dependent. Assuming no positive
jumps, we find concise formulas for these factors, and also characterize the
time of last exit from the negative half-line using three different approaches,
which further demonstrates applicability of path decomposition theory.
