Actual practical example of a prefix-free Turing-complete language
While not exactly what you want, esoteric languages Jot, Iota and Zot could be good starting points. They are all Turing complete.

In particular Iota language is defined as


syntax            semantic
F --&gt; i           ^x.xSK
F --&gt; *  F  F     [F][F]



So


  an Iota program is either an i, or a * followed by two Iota (sub-)programs


so it satisfies the prefix property. It's extension Zot adds input and output, where the input is the sequence of bits that follow the valid program description. By restricting Zot to programs with empty input you get a language that should satisfy your requirements. (Or alternatively adding just output to Iota.)

There is Haskell implementation of Zot.