The tacit knowledge description only talks about the acquisition process of the knowledge, not that the knowledge itself is outside bounds of rationality or the physical world. By the same token when you claim "humans have intelligence that is impossible to express through reason or codification" the entire argument hinges on the actual meaning of impossible. Is it impossible in itself, or ever, because human intelligence is non-reason based or are we using a meaning of impossible that at this day and age the process of doing that is still intractable for all intents and purposes. If you're claiming the former, that itself is such a strong claim it requires its own strong proof. If it's the latter, well we have been developing psychotechnologies for several millennia to be able to express ourselves and our cognitive processes and getting better at it, you just need to be patient.
If you found a working x86 chip in the wild, but all documentation and knowledgeable people were wiped out for some reason, I bet the process of finding out how that x86 worked would look quite similar. It wouldn't make x86 otherworldly or in need of a soul.