<p>I had experience with struts, and briefly experimented with tiles.  I know that within a layout each tile is a separate jsp page.  Suppose I hit a  button on each of four tiles, and each tile was (indirectly) calling a different web service.  Would the tiles refresh asynchronously as results came in at different times?</p>

I think it would depend on the code in the background.  You say that the tiles were indirectly calling web services.  I would think it would refresh asynchronously unless you specified in your code for it not to do so.  Say you push button 1 and then button 2.  Action 1 comes back and your code says that the screen needs to be updated and the page will refresh.  Once action 2 gets back, your code would again update the screen.  I believe this is how it works, but having never tried it, I can't be sure.
Why not just throw together a two tiled test page with buttons and try it out.  That should give you a definitive answer.