  In classical stochastic theory, the joint probability distributions of a
stochastic process obey by definition the Kolmogorov consistency conditions.
Interpreting such a process as a sequence of physical measurements with
probabilistic outcomes, these conditions reflect that the measurements do not
alter the state of the underlying physical system. Prominently, this assumption
has to be abandoned in the context of quantum mechanics, yet there are also
classical processes in which measurements influence the measured system. Here,
we derive conditions that characterize uniquely classical processes that are
probed by a reasonable class of invasive measurements. We then analyse under
what circumstances such classical processes can simulate the statistics arising
from quantum processes associated with informationally-complete measurements.
We expect that our investigation will help build a bridge between two
fundamental traits of non-classicality, namely, coherence and contextuality.
