There aren’t many options with built-in gaiters, because there isn’t much demand and it doesn’t work great in practice. A lot of people want gaiters only at certain times and like the option to add and remove them. Or they prefer to pick their favorite shoe and favorite gaiter rather than compromise on both.
That said, here are a couple. This one has a low cover that will keep low-cut socks dry but not much else.
And these are hard core for running over glaciers: