(coexisting on another dimensional plane)
The Tholian asteroid dock was a drydock facility constructed within an asteroid-moon that orbited a gas giant in the Vintaak system in the mirror universe. It was there that the Tholian Assembly, in 2155, harbored the USS Defiant, unexpectedly obtained from the future, for study and evaluation. (TOS: "The Tholian Web")
Unknown to the Assembly however, the Terran Empire became aware of the existence of the ship from the future, and sent forth ISS Enterprise to capture the ship. Though managing to destroy Enterprise in the ensuing battle, the Tholians were not able to prevent the capture of the Defiant, which, after coming under the control of a boarding party, managed to escape from their control and inflicted heavy damage on their facility, resulting in the structure exploding. (ENT: "In a Mirror, Darkly", "In a Mirror, Darkly, Part II")
The structure was not identified by name in the episode, nor in any other contemporary production source; its designer, Doug Drexler, coined the term "Tholian asteroid dock" after-the-fact. The Star Trek Encyclopedia (4th ed., vol. 2, p. 393) uses the term "Tholian asteroid base" for this structure.
The in-an-asteroid embedded Tholian spacedock for "In a Mirror, Darkly" nearly did not make it into the episode, as Producer Mike Sussman recalled; "I remember writing a description of the asteroid base in the script for part one of 'Mirror Darkly', half expecting it to get thrown out for budgetary or other reasons. And then at the production meeting Herman [Zimmerman] held up the image of the [Ralph] McQuarrie asteroid base and asked the producers if they wanted something like this. I got a chill down my spine, realizing the McQuarrie painting had probably planted the idea. It was just too cool that an unused concept painting for ST:TMP by the master himself was recycled something like a quarter century later at a production meeting for a Trek episode I wrote. Still gives me chills."
The Tholian spacedock was designed and built as a CGI model by Doug Drexler. Likewise, Drexler was inspired by pre-production paintings Ralph McQuarrie had done for an abandoned 1976-1977 project called Star Trek: Planet of the Titans (not Star Trek: The Motion Picture, as both Sussman and Drexler assumed). "I've always loved the idea of this, and was fascinated by Ralph McQuarrie's TMP asteroid base painting," Drexler stated.
Using pre-existing CGI parts from various other models, Drexler constructed his model in one day. "I had to do a variety of ideas in a couple of days so an afternoon is about it," he remembered. "Believe me, I had a kludge box full of parts and nernies. I couldn't afford to build it all from scratch. There are a lot of freighter parts in there!" Drexler sent off his model to Eden FX, where CGI modeler Mike Stetson expanded upon and fine-tuned the model, and where it was combined with the Constitution-class CG model of the USS Defiant, built by Koji Kuramura and Robert Bonchune for their appearance in the episode. Both models were re-used to represent the same scene in the 2006 reference book Ships of the Line (p. 29), and on the cover of the Star Trek: Enterprise Calendar (2006).