I don't know for certain, but it most likely has to do with the leftover sodium or chloride ions in the solution. One or another of those probably winds with dissolved silver ions in a way that ties up thiosulfate more than would be the case in a straight ammonium thiosulfate fixer. Adding sodium thiocyanate might help here (both in fixing speed and capacity), but that's very definitely not a supermarket chemical.
Instead of ammonium chloride, you can probably use ammonium bicarb (also known as baker's ammonia) which will give ammonium thio plus sodium carbonate. No chloride.
TF-4 is alkaline and Ilford Rapid is acidic. As for alkaline vs acidic fix, there are long discussions in here on that. Basically, it's easier to wash the fix and its by-products out of fiber paper if the fix is alkaline. By the way, welcome to APUG from another New Hampshire-ite! I...
Maybe not supermarket -- maybe Big Box store. Janitor in a Drum was weak HCl, but at home improvement stores it's sold as "muriatic acid" for cleaning concrete. Not sure what strength, however, which is somewhat critical in this case (might be sufficient to manage the solution pH if you have a pH meter).