The conventional wisdom was that with some Pyro formulations the stain can be intensified slightly in an alkaline after-bath. It didn't have to be the developer per se, just that the spent developer is a convenient alkaline solution that you'd have just used.
Originally Posted by jnanian
However based on everything I've read, including Anchell/Troop, if anything happens at all in the after-bath, the increase is to the general stain, not imagewise stain. So it's basically just a higher base fog level. No value. We also have to remember a lot of these traditions came about when films were quite different. With current films some of them stain to different degrees etc. Not all Pyro formulas are the same either. Stain color, intensity etc depends as much on the other components of the developing solution (the alkali etc) as it does on the developing agent. One notable example is Wimberley's WD2D+. It has a carbonate alkali and produces a yellow-orange stain as opposed to the more typical yellow-green or green-brown, and Wimberley clearly states that with this developer the full extent of tanning/staining is achieved in development.
Sandy King, Hutchings etc have all writen extensively on these developers, but in the end it is still somewhat murky and the only way to really know how a particular Pyro developer and film work together is to experiment, try a few things and see for yourself if stain intensifies in the wash, if an acid stop bath and/or fixer reduces stain, if a high level of sulfite after development (ie hypo clearing agent) reduces stain, etc etc.