Does piping MT19937 random stream into SHA512 make the state of MT unrecoverable in practice? It's well known that state of a Mersenne twister is quite easy to recover after you observe enough samples. But what happens if we pipe blocks comming out of MT into SHA512 and give this as output.
Since there is no mathematical proof that SHA512 is not invertible there probably wouldn't be a way to proove that MT state recovery in this scenario is impossible.

Is there any real practical way to obtain the MT state in this setup and hence be able to predict next blocks?
Or maybe prediction is somehow possible without state recovery?
I have read that there are CSPRNG designs that use an MT generator internally. Do they work like this, just take an MT stream and obfuscate it in some way or are they completely different?

Thank you!
The main issue would be the MT seed size. MT has a large enough state, but the seed is generally just a 32 bit word $w$. See here for more information.
SHA on the output won't guard you from a brute force attack on the seed; an attacker can just try and generate the stream and perform the SHA calculations and compare.
So you need to somehow extend the seed size and initialize the state. The state is pretty large and MT has a large period, so the plumbing is already there. If you can then just call it MT is the obvious next question.


Is there any real practical way to obtain the MT state in this setup and hence be able to predict next blocks?

Yes, as the initial state is derived from a generally small seed.

Or maybe prediction is somehow possible without state recovery?

I don't see that happening, but if you can retrieve the initial state, then you don't need to go there.

I have read that there are CSPRNG designs that use an MT generator internally. Do they work like this, just take an MT stream and obfuscate it in some way or are they completely different?

I haven't heard of them, but if they don't allow for a large seed as input, then they're in trouble.

The final question is indeed if you have gained anything when you're finished. Plenty of DRBG's based on hash functions exist, and adding a fast PRNG at the backend may not make much sense. The large state would not sit well with many cryptographers and secure software / hardware developers either.