I am having no end of trouble trying to find the magic to force a single-column RadTileView layout, with all tiles expanding to take up their fully available width.
What property should I be looking for which determines how the tiles determine their width?
The RadTileView control has properties which set the layout of all RadTileViewItems inside. For more information you can read this article. Also you can see some customizations implemented in our online demonstration projects. For your convenience I have implemented a one column layout of the RadTileView control in the attached project. Take a look at it and let me know if it works for you.