In practice, I have seen articles and financial textbooks on calibration of processes directly under the risk neutral world without showing that the measure is equivalent to a physical measure $P$. They seem to make an assumption that in the physical world, the market is arbitrage free, and that there exists an equivalent measure $P$, whatever the process defined on $Q$ looks like in $P$. Is there a reason for this and why people don't bother with even checking that there exists an equivalent measure $P$?
Fundamentally, option pricing is an extrapolation exercise. Fitting a q-measure model to the observed option prices gives a way of performing the extrapolation.
If q-measure model gives reasonable dynamics to the option prices observed and the underlying then the process of dynamic hedging with the options will work. If it doesn't, there will be systematic biases and the model will be poor.
Practitioners don't worry about the real-world process because it's unknowable and modelling it better rarely helps with the modelling and hedging.