This is not an answer to the OP's question, and is a bit of a tangent. But perhaps relevant concerning the physical realizability issue raised by Joel.

I just today heard a talk on a "Fold-and-Cut Machine," that showed that

"a fold-and-cut machine can decide a 3-SAT instance with $n$ variables and $m$ clauses using $O(nm+m^2)$ operations (...), showing that the machine is at least as powerful as a nondeterministic Turing machine."

An, Byoungkwon, Erik D. Demaine, Martin L. Demaine, and Jason S. Ku. "Computing 3SAT on a Fold-and-Cut Machine." Full CCCG Proceedings download, p.208ff for the article.

One folds a strip of paper a polynomial number of times, snips it to produce holes, rearranges the folding, and then looks to see if you can see all the way through the rearranged folding. You can iff the 3-SAT instance is solvable.

This is not an answer to the OP's question, and is a bit of a tangent. But perhaps relevant concerning the physical realizability issue raised by Joel.

I just today heard a talk on a "Fold-and-Cut Machine." This leads to a physical model equi-powerful to a nondeterministic Turing machine:

"a fold-and-cut machine can decide a 3-SAT instance with $n$ variables and $m$ clauses using $O(nm+m^2)$ operations (...), showing that the machine is at least as powerful as a nondeterministic Turing machine."

An, Byoungkwon, Erik D. Demaine, Martin L. Demaine, and Jason S. Ku. "Computing 3SAT on a Fold-and-Cut Machine." Full CCCG Proceedings download, p.208ff for the article.

One folds a strip of paper a polynomial number of times, snips it to produce holes, rearranges the folding, and then looks to see if you can see all the way through the rearranged folding. You can iff the 3-SAT instance is solvable.