So if we have a (classical) theory with fields $\phi$ and conjugate momenta $\phi$, we turn this into a (canonically quantised) quantum theory by promoting these to operators and impose some sort of commutation/anti-commutation rules, normally we inherit this from the Poisson bracket.
We then also have observables, which are functions of these field operators. Similarly, for conserved currents/charges, these are functions of these field operators. These are inspired/based off the classical expression, however, this doesn't tell us the ordering of the field operators in the observable - we can change the order classically but in QM changing the order gives changes of order $\hbar^2$.
However, when we use the path integral formalism, this ordering problem doesn't occur (even if we use Grassmann numbers you need contractions and hence observable quantities are bosonic).
So how do we know which canonical theory the path integral is equivalent to?
I guess a corollary question is: are the orders of operators in the observables in a canonically quantised theory unique, or are there many choices of ordering that could be valid (obviously they would be inequivalent and hence we could experimentally test all of the orderings and see what the universe actually obeys)?