It is possible to do, but would require some in depth coding, as well as an intimate knowledge of the workings of phpbb...
For now, what I (used to) do is to just open the report link in a new tab, and close that tab when finished making the report.
If (like me) you're used to using the 'middle click' to open links in new tabs, it is extremely easy.
(If you don't know how to do that, most scroll-wheel mice (and some non-scrolling) actually have a middle button, which can be clicked by pushing the scroll wheel straight down. Doing this 'middle click' with the cursor on a link is -- in every tabbed browser I've ever used -- a shortcut for opening that link in a new tab. Once you get used to it, you'll wonder how you ever lived without it.)
Brilliant! Now why did I not think of this myself? We're lucky to have such bright guys like you on the Board. Thanks for the tip.