Let`s say we have a wafer of compensated semiconductor material and another of the same semiconductor material with the same net doping level, but with only one sort of dopants, i.e. without compensation. Can we tell them apart by their behaviour (i.e. response to experiments) alone?
At least some of the compensated levels should be visible in mid-IR absorption or fluorescence spectroscopy.
Also if they retain a slight negative charge in the crystal, they should attract and localise positrons so they would manifest as a slightly slower decay in the positron annihilation spectroscopy (PAS) waveforms. This is somewhat involved and will not discriminate for all types of defects.
The contaminating elements, no matter if charged, will reduce electron/hole mobility, which will be observable when resistance is complemented by Hall measurement. This should be complemented by:
...changing the temperature or chemical environment, during which the balance of donors and acceptors can be broken and the dopants will manifest themselves in the temperature (etc.) dependent curves.
Note that an authoritative answer would require more detailed knowledge of the material under study. A quantitative measurement will also require a reference standard of pure semiconductor of the same kind.
Edit: Great thanks to F. Hájek for his professional advice. He also suggests that the level of compensation could be visible in optical luminescence: https://www.sciencedirect.com/science/article/pii/0038109878915867