The phase rule of silver complexation with thiosulfate ion includes the possibility of at least 6 silver compounds, if not more, and as many as 3 or more sodium salts of thiosulfate existing in solution or as solid materials (less soluable).
When ammonia is added, this number escalates dramatically due to the inclusion of the ammonium ion in the complexes that can form.
Fixing is one of the more complex chemical reactions in photography, but one of the easier to design working solutions for.
The tests described above work well enough to yield prints which keep well under a wide variety of conditions with no need for worry and no need for additional post fixation rinse solutions.
However, there are some interesting properties of fixation that have never been discussed outside of reports at EK that I know of, nor have many of the novel fixing agents that have been described in the literature been used even though they are quite useful.
From a practical standpoint, all of the technobabble aside, the use of a given fix packaged by a reputable manufacturer and the use of their published methodology will give exellent stability, archival for all practical purposes as evidenced by prints lasting for 100+ years. For even longer life, if desired, a selenium bath will top things off. Anything else is wasteful of time, energy and money.
Those old prints in your family album from 50+ years ago were not treated in clearing baths and etc....
PE