And what is this other ingredient if it's not something physical?

It could well be something physical, but going beyond mere computationality.

If I knew what it was, I'd be waiting for my Nobel prize now :)

It's kind of as Newton's physics was missing the secret ingredient of special relativity. It was still physical, not magical, but not discovered yet.

You seem to be arguing for conscience as the fifth force of nature, or something quite close to it. That's basically magic.

Only in the same sense as dark matter or string theory are "basically magic". It is naive to assume we've already discovered all properties of matter and energy and are now only refining our understanding of those.

Yes, astrology could also work due to a fifth force of nature. After all, it would be naive to just dismiss it based on our current understanding of Physics.

It's not a good analogy. Astrology is an explanation of a problem that doesn't exist in the first place.