Has anyone ever experimented with an "unbalun" transformer, where the internal windings might be appropriately out of balance to feed an out of balance antenna, (off center fed)?
Sure - normally used with non-resonant verticals, or other non-balanced wire setups. Balanced antennas in the real world are usually not always balanced at all due to the surrounding environment, so an "UN-UN" might be a better choice.
While you are awaiting for broadband ferrites to arrive, you can cook up something old-school with air-wound balun / unun on say a 1-inch PVC pipe.
About half-way down the page this project has a nice schematic and wiring diagram of a typical pre-ferrite era 4:1 air wound:
SM2YER Goran's Homepage
Essentially, 13 bifilar wound turns on 1-inch or so pvc. Wired as shown it is a 4:1 balun.
If you want to turn it into an UN-UN, swap the center conductor and shield of the coax connector - keep the wires the same. In the case of a non-resonant vertical wired with an un-un, the shield connection side of the unun would also attach to radials. The other free wire of the coil merely goes to your antenna wire. Follow this with a good 1:1 choke.
The major problem with air-core bal / ununs, like the coax-wound "ugly" baluns, is that they are narrow-banded, and also have considerable reactance and leakage current. In this case, much capacitive reactance at 10m, which is usually problematic.
Most people overestimate the coverage. The following is a bit more reasonable.
13 bifilar turns (26 individual turns total) = 13mhz to 21mhz <--- typically overestimated as 160-10m in most articles...
18 turns = 3.5 to 7mhz <--- low band coverage
10 turns = 18 to 28mhz <-- higher band coverage, but still, capacitive reactance can be fiddly.
Still, 10m is a bit problematic with air-core, but if your application isn't too critical (perhaps an rx-only setup) then you may not notice a big difference.
So, knowing that these are narrowband devices, with some funky reactance and leakage current, there is no wonder that ferrites became so popular.
Fun to whip up on a whim to try out. Sometimes we over-engineer things depending on the application.