Anti-aliasing has to gradate between the foreground (text) color and the background behind it or it will have a nasty 'halo'. In this instant, the text is crossing two slightly different backgrounds so they have to be combined into one composite image.
I made a rectangular selection containing the text and its two backgrounds and 'copy merged' it into a new document. What I wanted was just the text on a transparent background but with an anti-aliased fringe derived from the background layers.
I used the 'magic wand' tool in Photoshop set to a low tolerance to select the text.
The selection was expanded by 1 pixel to include just a little of the background.
Then the selection was inversed to select everything but the text ...
... and cleared leaving just the anti-aliased text with the correct colors surrounding it. The image was finally saved as a 32-color, transparent GIF.
Now, when it is placed on the final Web page across two background images, it blends into them seamlessly.