As can for example be learned from chapter I.2 of Anthony Zee's Quantum field theory in a nutshell, path integrals can be used to to calculate the amplitude for a system to transition from one state to another by considering all possible paths between the two states.
This article gives a nice technical introduction to consistent histories and explains that this method can be used to calculate the answer to questions about alternative histories in a consistent set. For two histories to be consistent, their probabilities must be mutually exclusive such that their probabilities are "additive".
Looking at these two methods to calculate answers to meaningful questions in quantum mechanics (and how the relevant formulas are derived), they seem very similar to me.
So my question is:
What is the exact relationship between the method of path integral and consistent histories? Could one say that the consistent histories are some kind of the "classical limit" of the path integrals? Or does coarse graining (by an appropriate method?) Feynman's path integrals lead to consistent histories as a limit (IR fixed point)?
(In the article about the consistent histories it is mentioned, that if one considers too fine grained histories, they are due to the uncertainty principle no longer consistent and would resemble Feynman's path integrals. But I'd like to see an extended technical / mathematical explanation of the relationship between the two things to really understand it.)