The houses themselves do not have bases/homes/what-have-you outside of Shard, at least currently - and I don't see it happening in the near future. Not to say that it could never be justified in any way, but I personally see it as fairly tied to Shard.

Elotheans raised outside of Shard and its environs likely wouldn't formally belong to a house without making the trek back to intentionally join, with the possible exception of children born to house members outside of Shard - according to the book, for all house members except Waxen Moons ("Unlike all the other houses, a child born of the House is not necessarily a member."), though that doesn't specifically state that the membership is automatically extended at birth, vs. some later age when the child chooses to take up their membership vs. pursuing membership in a different house. Even then, how is the house getting that info? Are they sending a birth announcement or something? Would the house demand presentation of the child for some ceremonial recognition? None of THAT is said in the book, and if you think of the widespread use of christening-like ceremonies, it wouldn't be hard to imagine that the houses would have something similar for babies born to members. Still, while the houses are a distinct part of the Elothean culture, especially in Shard, an Elothean raised outside of that culture may not see the cultural stigma/pity toward a senka'i neir that they would if they were in Shard, since, well, they're surrounded by people who aren't part of the houses.

And one could easily imagine that the stigma may have lessened over time. Just because it was there when the book was said to have been written, doesn't mean it's necessarily just as strongly present now. You could recognize the time during which there was no way for Elothean players to have their characters officially join a house as a time that would reflect the lessened pity for the senka'i neir. With the occupation and the retaking of Shard, there'd be an understandable resurgence in patriotism/cultural identity and pride, and thus a restrengthening of those house ties (coinciding nicely with the availability of a way for the Elothean players to have their characters officially join houses), and perhaps the pity for the senka'i neir would have likewise restrengthened - or perhaps it would have been largely left behind, as the house Elotheans may even prefer not to use the somewhat derogatory term toward their fellow Elotheans as they all work together to rebuild (cue waving image of the Shard crest and playing of fanfare, chants of I-li-thi! I-li-thi!, etc.).

Those are my PERSONAL views (except for the quote taken from the book), though - Abasha may feel a bit differently, and I don't recall that there's anything formally said about the houses' existence outside of Shard. Even if there IS something stated officially for the time of the book, the scattering of Elotheans during the occupation of Shard may have changed that, as there would then have been an influx of housed Elotheans in various areas that may have lead to the establishment of some informal gatherings elsewhere - not any formal building, but rather, a monthly tea-at-Vaeldriil's or what-have-you, that may or may not be specific to any one Elothean house (given that there may have been a handful of members of this one, three of that one, etc., and more of a desire toward encouraging unity among the displaced Elotheans in an area).

-V., rambling as usual.