  We investigate the role of coherence and Markovianity in finding an answer to
the question whether the outcomes of a projectively measured quantum stochastic
process are compatible with a classical stochastic process. For this purpose we
put forward an operationally motivated definition of incoherent dynamics
applicable to any open system's dynamics. For non-degenerate observables
described by rank-1 projective measurements we show that classicality always
implies incoherent dynamics, whereas the converse is only true for invertible
Markovian (but not necessarily time-homogeneous) dynamics. For degenerate
observables the picture is somewhat reversed as classicality does no longer
suffice to imply incoherent dynamics (even in the invertible Markovian case),
while an incoherent, invertible Markovian dynamics still implies classicality.
