It is really, really clear what my anti-aliasing is doing when looking at the plane fading out into the distance. Below are some images (no shading - it is easier to see the plane approach the horizon without any shading) of different sample rates. Getting the plane to work took a bit of trial and error, but I think it is looking OK.
I used a completely square image that tiles perfectly. Here's the tile pattern I used (thanks Nate Gauntt):