  We analyze the multitime statistics associated with pure dephasing systems
repeatedly probed with sharp measurements, and search for measurement protocols
whose statistics satisfies the Kolmogorov consistency conditions possibly up to
a finite order. We find a rich phenomenology of quantum dephasing processes
which can be interpreted in classical terms. In particular, if the underlying
dephasing process is Markovian, we find sufficient conditions under which
classicality at every order can be found: this can be reached by choosing the
dephasing and measurement basis to be fully compatible or fully incompatible,
that is, mutually unbiased bases (MUBs). For non-Markovian processes,
classicality can only be proven in the fully compatible case, thus revealing a
key difference between Markovian and non-Markovian pure dephasing processes.
