Universal binary rewriting system What is the simplest example of a rewriting system from binary strings to binary strings

$$f:\Sigma^*\rightarrow\Sigma^*\qquad\Sigma=\{0,1\}$$

that can perform universal computation? Binary string rewriting systems in general can compute any computable function, but I have trouble finding particular instances that can by themselves compute any computable function given an appropriate input.  I've seen statements that a class of rewriting systems (e.g., the set of cyclic tag systems) is Turing-complete, but I'm looking for a single rewriting system that is universal.

I was thinking a self-modifying bitwise cyclic tag system might be a candidate, but I'm not sure how to interpret the output of such a system.
Rule 110 is a binary rewriting system that can perform universal computation, i.e., it has been proven to be universal.  It can be implemented by a finite-state transducer: it needs only finite state.

However, Rule 110 is not a tag system or a cyclic tag system, so this does not provide an instance of a specific binary tag system that is known to be universal.  It might be that examining the proof of universality of Rule 110 could yield such a system, as apparently the proof involves a reduction that goes by way of cyclic tag systems -- though personally I've never read the proof, so this is only speculation.

A side note: From Rule 110, you can construct a particular queue automaton that is universal: the queue alphabet is $\{0,1,\$\}$ and contains the state of the cellular automaton (a binary string representing the contents of each cell, followed by $\$$).  I don't know whether it'd be possible to use this to construct a specific tag system that is universal (e.g., if you can find a way to use a tag system to emulate a queue automaton).