I don't know your camera/monitor model nor how it is wired. The comments below are based on my installation of several LCD systems purchased from ebay. I am also assuming that you have only one camera connected to the monitor.
On most systems, there is a separate power feed from the reverse lighting circuit (backup lights) that cause power to be applied to the monitor/camera if the power is off to the camera/monitor. If the camera/monitor is already powered, switching to reverse should keep the camera/monitor on. In my opinion, something is wrong in the camera/monitor switching.
Since you desire to have it on all the time, try disconnecting the power feeds one at the time. One power feed will be on when the ignition is powered, and the other power feed will only be on when the backup lights are illuminated. If you can identify and disconnect the power feed from the backup light circuit, I believe your problem will be solved.
You can try to use a google search with your monitor model number to find a wiring diagram that may help you identify the correct wire.
Let us know the final outcome.
Fred & Vicki
Richmond Hill, Ga
2000 H.R.Endeavor/Freightliner/330 Cat