Ubuntu – Virtual VNC Screen/Workspace

That's what DMX: Distributed Multi-head X is designed to do, unfortunately it's been broken in several X.Org releases, so you may have to work a bit to get it working.

The trick here is to generate a virtual monitor on the server, place it in relation to the real monitor wherever you would like it to be, then instruct VNC to use that portion of the total monitor (real + virtual) you would like to use. All of this is well documented on the ever helpful Arch Linux forum.

1. To create a virtual mirror:

   $ gtf 1920 1080 60
   # 1920x1080 @ 60.00 Hz (GTF) hsync: 67.08 kHz; pclk: 172.80 MHz
   Modeline "1920x1080_60.00"  172.80  1920 2040 2248 2576  1080 1081 1084 1118  -HSync +Vsync
   

   allows you to find the modeline needed: this assumes a virtual monitor of size 1920x1080, and a refresh rate rate of 60Hz, you may adjust this to your needs.

   Now you may generate the new modeline by means of

   xrandr --newmode "1920x1080_60.00"  172.80  1920 2040 2248 2576  1080 1081 1084 1118  -HSync +Vsync
   

2. You may now generate the VIRTUAL1 monitor by means of:

   xrandr --addmode VIRTUAL1 1920x1080_60.00
   

3. Now you may place the virtual monitor to the left of your existing monitor (HDMI1, in my case, change this as it best suits you):

   xrandr --output VIRTUAL1 --mode 1920x1080_60.00 --left-of HDMI1
   

4. Lastly, you may decide to see only the virtual part of the display (for instance) by means of

   x11vnc -clip 1920x1080+0+0
   x11vnc -clip xinerama1
   

   (whichever works for you). If you want to see the whole monitor space (virtual + real), just omit the above.

5. The original post referenced above suggests passing the following two parameters on the remote computer, when you wish to start the vnc session:

   vncviewer -encodings "tight copyrect"
   

   I never found this necessary, but I will pass it along to you since YMMV.

P.S.: your description of how X11 should treat the presence of the two distinct monitors is confusing. What X11 does easily, and without any need of our intervention, is to create a single workspace (called a screen) out of the existing monitors, in such a way that windows can be dragged from one monitor to another, and the mouse flows without barriers over the whole monitor space (the screen). Also, maximizing windows fills only the monitor they are located in, not the screen. This is what I assumed you desired, and what the above achieves without any extra work. Anything else requires work (if it can be done at all).