The following seems to do the trick!
private function onReplayClick(e:MouseEvent):void {
var theStage:Stage = this.stage; //Temporarly store the stage.
//Kill any animations happening:
TweenMax.killAll(); //3rd party, may not be applicable for you :P
//Remove ALL containers / child DisplayObjects
SpriteUtils.recursiveRemove(theStage); //custom-made
// (this object is no longer attached to the stage at this point)
//Nullify any Singleton / Static variables:
Main.INST = null;
// Load the 'bytes' of the current SWF in a new Loader, then...
// add it to the stage
var swf:Loader = new Loader();
swf.loadBytes( theStage.loaderInfo.bytes );
theStage.addChild( swf );
}
By doing a deep recursive cleanup of the DisplayObjects, and any static variables (like Singleton instances), it leaves you with a blank stage.
After that, you can instantiate a new Loader that will load the SWF itself via the current LoaderInfo's bytes property.
Add the Loader to the Stage, and you're up and running again!