The Illithids maintain a presence in Bralspace, in part the thwart any ideas of the IEN and also due to recent events during the UWII. They have not, however, intervened on either side in this conflict. However, given the proximity of Bralspace to the sphere containing their homeworld (Falx), they are most certainly interested in ensuring that Bralspace is not controlled by hostile forces.
Instead, Bralspace is the patrol centre of the 2nd Illithid Fleet which at any given time is comprised of up to 9 Nautiloids and 2 Dreadnoughts. They occasionally send one to dock at Bral to check in with the Embassy there, which is also where they get news and orders from Central Command on Falx (via extradimensional means such as Probability Travel, a psionic discipline possessed by very talented Mind Flayers, who tend to get the "ambassador" job!). However, the main base of operations (as well as a holding area for weapons, supplies and slaves) is hidden on a base deep within the Gods' Marbles.
(The Rock of Bral)
The Illithid Nation maintain an embassy on Bral itself (see Bral map included in the original boxed set) which they use to keep an eye on the local happenings. As Bral is a popular waystation for most spacefaring species, this makes it an excellent spot for spy operations. The Illithid do not OPENLY trade in slaves, though rumours persist that such a trade is going on, with deals made at Bral while the actual "prisoner exchanges" take place near The Gods' Marbles. The other party in such deals are unlikely to be Neogi, but include humanoid races (human slavers, drow perhaps) and perhaps even Beholders.
Meanwhile, the inhabitants of Bral regard the Illithids as a stable presence, as they trade extensively on many above board items such as precious gems, metals and magic items. Note that Falx, their homeworld sphere, is just down the Arcane Inner flow from Bral (perhaps 20-30 days), so they are very much concerned over what happens in this sphere.
The Embassy at Bral is also the check-in point for any Illithid Navy vessels which make port at Bral. There always seems to be at least 1 Nautiloid docked at Bral, suggesting they maintain a sizeable fleet close by "just in case". An occasional Dreadnought has also been spotted docked at the station. Some interpret this as a none-too-subtle message to any would-be invaders of Bral that they are around and watching closely! Some even suggest that Prince Andru has made some sort of deal or alliance with the Illithids in exchange for this protection.
(Gods' Marbles)
Hidden within the asteroid chain known as Gods' Marbles, the Illithids have established a secret naval base. Basically, this is a small spherical planetoid (Size A Rock world, 1 km dia) which they have hollowed out over the centuries. The outside is airless and cratered. At the base of several craters are circular openings, typically 100-300 ft across, which lead to cylindrical passage that allow a SJing vessel to travel into the interior. There are 10 such openings, arrayed in a random fashion over the entire surface. Each is camouflaged/concealed and hence hard to find unless you get very close to the surface. Some of the larger ones have Hallucinatory Terrain cast over them to conceal their location. True Seeing or similar will reveal these clearly, which is how the Illithids navigate to the location precisely.
A sentry of at least 3 Nautiloids is maintained nearby, which will attempt to destroy any intruder who wanders too close to the base. They will typically avoid tangling with an obviously superior force, instead trying to flee to Bral, but will send a message to the base to defend itself just in case. They will NOT, however, reveal its location by fleeing into it, or flying too close, or anything similarly stupid! Instead, the picket vessels will endeavour to battle such opponents a fair distance away from the base.
The entrance to each tunnel is defended by a gun battery, concealed in the lip that surrounds it. Typically, there will be 2-5 ballista and one heavier weapon (catapult, bombard, heavy jettison). If desperate, the defenders can also activate a winch system that pulls a number of iron chains across the opening (similar to harbour nets used to guard terrestrial harbour openings against submarine incursions) to prevent an enemy ship from passing through without being cut up/fowled by them!
Within the base, the Illithids have established a roughly spherical area similar to that found within a Dyson sphere. The wall is quite thick, so the distance along one of the above passage to the interior is typically 800-1000ft. The interior is filled with air, and the gravity plane coincides with the inner surface, and gets progressively weaker towards the geometric centre of the world. At this location floats a massive docking structure which is moored in place by a large number of rope bridges, catwalks, etc. That extend to the inside surface where the inhabitants live. SJing vessels dock at this location (only Illithid navy ships!), and visitors walk "down" towards the surface. In most cases there are mechanised elevator/escalator devices or trolley systems, similar to those one would see at a ski hill, to assist in the unusual gravity transits required.
At any given time, there will be 1-4 Nautiloids, 0-1 Dreadnoughts and several smaller vessels docked at the station.
The inner surface of the world is home to the naval base itself, which has a typical population of 1000 Illithids and double that number of slaves. It is similar in construction and style to the great cities on Falx, though much smaller. There are a few areas that have been converted into Fungus Farms to grow food used to feed the slaves (and the Illithids themselves, though they prefer the imported stuff or occasional slave brain). The fungus grown includes a species that emits low levels of purple (ultraviolet) light, which is ideal for Illithid sight, though difficult to see by normal humanoids. The entire inside world has a very surreal look to it as seen by humanoid eyes - similar to the effect of a black light in a dark room.
A few shallow lakes and swampy areas have been created, too, though this world is far too small to generate any sort of weather systems. These pools are used for breeding purposes (and drinking water) and also used to raise the immature tadpoles used in some Illithid helms.
There are large storage facilities and slave quarters throughout, all heavily guarded. The base maintains a constant oversupply (up to 3 months worth) of critical items in case of blockade or war in the sphere. The base also has repair facilities for Illithid vessels (including helms), located within the central docking facility or "Hub".
The command centre is down on the surface, and includes a large pool device which acts as a massive planetary locator for Bralspace. This locator is able to track all objects of reasonable size, including SJing vessels over 25T, and is used by the Illithids to keep track (literally) of what's going on. The command centre commander is always a very powerful psionic mind flayer who is able to "converse" with his counterpart at the Illithid Embassy on Bral by telepathic means. He is also able to exchange thoughts with other mind flayers in the sphere, as long as they are strapped into an Illithid Series helm (which uses psionic energy). This affords the Central Commander the ability to monitor and co-ordinate the elements of the 2nd Fleet that have been made available to him in this sphere.