The only software I know that can re-broadcast RealMedia RTSP stream is VLC. However depending on a streaming server you can send not only Flash video, but also MPEG-2 video or video of some other formats to the streaming server that will broadcast RTMP flash stream.

But there could be many other solutions. Say you can send the video from your graphics card to a digital video camera and then use the video from the digital camera on a streaming server.

Are you sure that they re-broadcast RTSP Real Media streaming video?