"Our house has a pool" for example. There is a very specific pool being referred to, but it's sufficiently indirect that "the" would never be used here without radically changing sentence meaning.
It's like saying "our house has a roof" — very similar, but nobody really cares about the roof, the focus remains on the house.
If you refer to that same pool without introducing it, it will still be marked for definiteness - compare my pool, my house's pool, or the pool at my house. But your example sentence doesn't work that way.
(Note that the three phrases I listed are all bare noun phrases that can't stand as sentences. Note also that if you used one of those phrases when your listener wasn't already aware that you had a pool, they might feel that you "moved too fast" in the conversation, assuming that they knew about your pool when they didn't.)