My map, which has an opaque white background, shows the following when being tiled at its edge:

It's a bit hard to tell, but this is a square image with the white background of the map's right edge, and the black is the background that is automatically added by TileUp when in order to make the image square.

Could we have some option to specify that color and/or to make it transparent if exporting in a file format that supports transparency (e.g. PNG)?