I appear to have found some clear acrylic or polycarbonate globes of various sizes on Ebay listed under "HEAVY REPLACEMENT LIGHT GLOBES & POST TOP FIXTURES", and you can get them with screw-on lipped bases. They come in several sizes including the following:
12" diameter - $22 = 3.9 gallons
16" diameter - $44 = 9.3 gallons
20" diameter - $95 = 18 gallons
I'd probably opt for the 12", which has a screw-on base available from the save vendor for $12, but might opt for one of the larger if I decided to try this with anything apart from Hawaiian Volcano Shrimp.
They can also be bought with an opening which does not have a lip, which could allow for a very minimal design if it can sit on a counter alone without any additional base, but that could be difficult then to seal it. Simply a circular disk of acrylic could be welded with acrylic glue to the bottom, so long as you glued around the outside with the piece already in place to not get any fumes trapped inside.
But a threaded lipped base might be easier, especially in case you need to open the sphere up to make any adjustments. And it might thus be able to be sealed with silicon instead, as acrylic sealant is very permanent stuff. A base would allow me to use an LED chip underneath which could be a significant advantage, and maybe I'd be better off without any substrate even.
The volcano shrimp are hardy enough to not be bothered by temperature fluctuations it seems, while if I had say cherry shrimp then I might need to forego the cordless simplicity of it by including say a hydor heating pad heater under the base, despite that they're not quite ideal as they lack a thermostat. However, volcano shrimp require brackish water, which would somewhat limit me if I was considering including plants.
These plant species can apparently still succeed in brackish conditions though - Anacharis, Anubias Nanas, Aquatic Banana plants, Dwarf Sagittaria Subulata, potted Narrow Leaf Chain Swords, Bacopa, Cabomba Green, Water Sprite, Temple plant, Crypt Ciliata, Java Fern, Java Moss, Tall Sagittaria Subulata, Jungle Vals, and Duckweed.
Do people think plants might work in this or not? And I still have no idea whether it would be better to start lean or to start with a nutrient-rich environment though. I could add trace nutrients, but still run a slightly nitrogen-poor environment in it. Plants would show suffering it time though. And I'm not sure if trace nutrients would continue to be recycled or whether they'd be depleted eventually by being bound into other compounds which the system cannot readily utilize, though I suspect the former.
I do need to look into cherry/crystal shrimp to find out if there is some other food I could naturally grow in such an environment for them to eat, such as the suggested 'bio-film' for instance.