A complex interaction between:
1) parasite drag - air friction on the overall airframe, which
decreases in thinner air for a given airframe at a given true speed.
2) induced (lift) drag - which
increases in thinner air for a given airframe at a given true speed. Higher AoA required in thinner air.
Meaning that in TAS, Vy (the most efficient climb speed) increases with altitude - you can climb best at some higher TAS, as you can reduce AoA (and induced drag) without incurring as big a penalty from parasite drag.
https://www.boldmethod.com/learn-to-...ere-they-meet/
3) IAS gauge/pitot/static arrangement, which can introduce measurement errors.
While there is a defined
theoretical relationship between IAS and TAS divergence as altitude increases, that relationship is only approximate when applied to a real-world aircraft: where the pitot tube is
located, how well the pitot tube is
aligned with the relative wind/slipstream at various pitch/climb angles, etc. etc. For a specific airframe, that can be corrected to calibrated airspeed (CAS), but that is not on "the big gauge" that pilots generally watch - one has to use tables, or look at some fine print somewhere in the cockpit. And the CAS/IAS relationship is not linear - CAS may be 2 knots slower than IAS at 140 KIAS, and 1 knot slower at 100 KIAS, and 2 knots faster at 60 KIAS.
Vy in CAS could well be identical at 0 feet and 20000 feet - but the gauge won't tell you your CAS.
4) Ultimately, what goes into the manuals is results from test flights, rather than theory. A Mooney test pilot took 'er up to 20000 feet, and experimented, and reported that up there, (s)he got the best ROC at 93 kts IAS rather than 89 kts - in that airframe design. And they put that in the book.
Outside of the fact that a turbo allows one to fly across a larger altitude and ambient density range, I'm not sure a turbo has any
direct effect on rated Vy, and changes therein. One would have to experiment with a normally-aspirated version of an identical airframe for comparison.
Now, in a
twin, the fact that the props "blow the wing" and add airflow and lift, might mean a noticable change in Vy with/without turbocharging (more power).