Apparently bamboo can also be processed into "bamboo linen", i.e. using the same process as with flax or hemp. One commercial producer of bamboo linen is Litrax. Its bamboo linens are described/advertised on textileworld.com. The processing, however, uses "fine-tuned enzymatic cocktails", as is to be expected.
At this page, more information on the method used by Litrax can be found. From the same page:
the typical rayon production process used for most manufactured fiber derived from bamboo uses caustic chemicals whose emissions can harm not only factory workers but also the environment when not properly controlled within the factory. And the resulting fiber — whether processed using the rayon process or a closed-loop, eco-friendly, modified lyocell process — does not resemble bamboo at all in its original form. In addition, widespread claims that the fiber offers antimicrobial properties cannot be substantiated, and its biodegradability has been questioned because it does not break down within a reasonably short time after disposal in a landfill or in a recycling scenario.
However, Litrax seems to produce only yarns and fibres, and doesn't sell any clothing itself. In my case, I am looking for something that can be purchasable commercially, but can't find any sources.
Specifically I'm most interested in availability to the UK, including online.