You can export it as a QuickTime movie and choose the Manual Advance Playback mode. It will behave exactly as a Presentation (click to advance and so on) and you will not loose any of the effects (animations, transitions, etc).
Once you have it in Quicktime then you can just present it in a window. Make sure you select Full Quality so you get the best possible movie out.
Make sure you have "Allow Exposé, Dashboard and others to use the screen" in keynote preferences under Slideshow.
Then, before you slide over to Safari, press the "F" key to pause the slideshow. This prevents the slideshow from disappearing.
You can then slide over, or use Mission Control, etc without affecting the slideshow.
There is a bug here, to go back to the slideshow you have to slide back to it or click on the desktop in Mission Control; you can't click the Dock icon or Cmd-Tab back to it.
(Clicking the dock icon / cmd-tabbing will just hide the dock and menubar but not take you back to the presentation)
Also: When you get back to the slideshow, the first key press or mouse click unpauses the presentation, so you will have to click / press again to advance to the next slide.
Side note: instead of having keynote and Safari on separate desktops, you can have them both as fullscreen apps. This hides the Dock and Menubar, so when you show the web page, there is more room for it.
Best Answer
I had a similar problem and found a surprisingly simple workaround on the Apple Support Forum (https://discussions.apple.com/thread/5522746?tstart=0)
I needed a way to show my presentation on a webinar and still have access to the webinar control panel so I can see questions when they come in without having to close the presentation first.
The SOLUTION: You basically just need to export your completed Keynote presentation as an HTML file and open it in a browser. Then simply make it full screen on the screen or projector you're showing it on.
I used Chrome and found that the address bar still appeared in full screen view. To fix this I simple unchecked the "Always show toolbar in full screen" option in Chrome's View menu.
Solved the problem for me. Transitions worked too.
I know this thread is pretty old but I hope that helps someone searching for help with now (like I was)