I finally found an article that doesn't just assume that each time you have a baby there's a 50/50 (or technically 51/49) chance that you'll have the same gender again. This is a tougher search problem than it sounds, because so many science explainers take the easy way out and refer to pregnancy as a coin flip. Obviously the previous coin flips don't inform later ones. But how do you know birth is a coin flip? I suggest that the immune system + reproductive system + parental choice + environmental factors = something more complicated than a coin flip, something with memory potential. SO ... I finally found this article, which references an earlier one done by actual statisticians:
It turns out there may be a slightly increased chance (up 2 to 6%) of having a fourth boy if you already have three. Maybe not the same for girls, interestingly. I'd guess if we have another child, there's a ~55% chance of having a boy, based just on this survey of actual family birth orders. And that sounds about right: nothing too different from the "coin flip" explanation, but something revealing a system that's more complex and interesting. Personally, I wonder if my NKG2D or MICA immunoproteins might not be involved?