From Monte-Carlo simulation Hull-White process I get paths in risk-neutal measure. How can I get paths in physical measure?

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It is not easy to move from risk-neutral to real-world probabilities as this would involve estimating the market risk permium.

The easiest would be to use two separate calibrations:

You need do the market price calibration. As the previous person answered, you can use likelihood to check. To certain limitation of the model itself (in this case HW), even you find good fit parameters, the physical path may not be that ideal or you can never achieve. Not sure if this answer your question but trying to be helpful.

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