Here is a diagram of the system in question (adapted from the diagram on this page):
All of the work produced by the Carnot engine is used to drive the Carnot refrigerator, which is simply a Carnot engine operating in reverse. Accordingly, no work is performed outside the system. Despite the fact that no work is performed, such a system would create a perpetual flow of heat from the hot reservoir, through the Carnot engine, through the cold reservoir, and back up through the Carnot refrigerator. Also, because the Carnot engine is driving another Carnot engine in reverse, the net entropy increase is zero, ie all the heat that flows from the hot reservoir to the cold one to drive the Carnot engine is pumped back to the hot reservoir by the Carnot refrigerator.
I realize that such a perpetual flow of heat is only theoretical. Any physical realization would lose heat due to friction. I'm also aware that my question seems similar to a naive attempt at perpetual motion, but that is not at all the focus of my question. I am trying carefully distinguish between a flow of heat that does no work and any motion that produces work.
The focus of my question is simply whether such a theoretical system would indeed create a perpetual flow of energy with no increase in entropy. If so, this would mean that while perpetual motion (aka perpetual work) is theoretically impossible, perpetual flow (that does no work) is possible.
I'm also curious as to why such a system has not been discussed in the thermodynamic literature, even if merely to dismiss it as a possibility. I searched pretty extensively (using both Google and Google Scholar) and found no discussion of such a system, which is why I am now asking here. The only related discussion concerned a similar system that tried to bleed off some of the work produced by the Carnot engine to drive an external system. Such a configuration is clearly an attempt at a perpetual motion machine.