One thing we don't know is whether the brain is maybe able to perform calculations on real numbers which could allow for different or even more powerful calculations compared to the Turing machine.

No.

The line of thinking is, "Well, if we had real numbers, we could compute uncomputable functions." No shit. Most real numbers are not computable. If you start with something not computable, you end up with something not computable.

The big problem is that there is no way you will get more than several digits of precision on anything. The next problem is that there's zero evidence that anything in the universe is capable of encoding an arbitrary real number, even within a given bound. The last problem is that making a "real-number" machine capable of solving a problem that a Turing machine can't basically requires knowing the answer ahead of time and encoding it into the real-number machine as a constant. Unclear how you'd do that.

How I understood it, and please correct me if i am wrong, is that a machine than can caluclate numbers with infinite precision in constant time has certainly a time benefit, but not necessarily a broader class of computable functions compared to a Turing machine. I interpreted z3phyr’s reference to Wikipedia as a hint to that idea. But I agree that the most convincing arguments are against the existence of such a machine.

(By the way, writing "No.\n\n" can be received as a rude response. I felt a little bit uncomfortable reading it.)

Oops, sorry for this comment. ("don't drink and internet")

It is extremely unlikely due to noise. It's the same reason we use digital computers right now... for all the "waste" of converting voltages into 1 and 0 it pays back in spades in that we get reliable computing. You hit the noise floor of an analog system long before you're into any sort of revolutionary computation land. It's not much use if something can theoretically compute something amazing if it can't do any of receive that degree of input, maintain that degree of accuracy during the computation, or output it to that degree of accuracy.