OK here's an update at day 346.
A few months ago, the setup experienced a very robust growth of dark green Cyanobacteria. Unfortunately, I did not get any photographs during this period. While the microbial mat was impressively thick (~4mm), it was difficult to keep the tank looking very good because in addition to growing lots of Cyanobacteria on the stone top surfaces, it also grew all over the sand and glass very fast.
I suspect there is a close relationship between the relative concentrations of nitrate and phosphate and the degree to which the different photosynthesizes proliferate. The stones are currently growing a microbial mat dominated by dark amber-colored benthic Diatoms with relatively little green Cyanobacteria coverage. The Diatoms are more well- mannered and grow only on the stones.
I've been adding the Brightwell Aquatics Reef Code A + B for calcium and alkilinity in the hope that it will boost CaCO3 accretion under the microbial mat. I haven't been monitoring very carefully though aside from occasional Ca tests with the API kit. I'm not a chemist and I don't know what effect this very salty water (85ppt) might have on titrant tests formulated for regular seawater(?). At some point I'm going to figure out a more rigorous approach and more carefully measure dissolved nutrients and other parameters.
Presumably there are heterotrophs, sulfate reducers and more in stratified zones between the Cyanobacteria and Diatoms extracellular polymeric substances (EPS) and the stone top surfaces. The interactions between these different organisms and the diurnal light-darkness cycle is what leads to formation of calcium carbonate and entrapment of sediment particles in stromatolite layers. I have sampled the stones here and I there is indeed a hardened layer on the top surfaces. It is not adhered very well, however, and I can easily scrape it off with my fingernail to expose the very white limestone beneath. I hope that with some more experimentation and maintenance of favorable conditions I will eventually observe more robust layer accretion and stromatolite growth.
Both the Diatoms and the Cyanobacteria produce considerable free-floating exudate. For a more appealing aquarium display, I need to beef up the filtration to reduce the surface film, floating foam and water discoloration. This little Aqueon internal filter works OK, although it's pretty puny and doesn't have enough flow for this tank. Sometimes I run a UV filter inside and sometimes I just use the filter pad. I might get another one of these filters, but in a bigger size.
I've also considered a protein skimmer, although I wonder if operating one would export too many nutrients and starve the photosynthesizes. I could build one of this kind myself but shorter to suit this shallow tank...
Bulk Reef Supply: Nano Protein Skimmer - Reef Glass