There are monitors that natively support SOG (sync-on-green), and others that don't, the others that don't support it, they need some extra components to separate the video sync from the green channel, and that can be done with a LM1881 chip in order to make it work.
In this example, i was modding my Playstation 2 to support VGA output, but PS2 has a SOG output and i ended up with a greenish color on my new HP Pavilion 22xw monitor, although everything was correctly connected.
The reason is simple, the LM1881 can separate the sync from the green channel, but the synchronization waves are still present inside the green channel that connects to your monitor (because LM1881 doesn't have a sync removal from green, so you need to connect green to green directly), and some monitors interpret those waves as color, instead of interpreting it as synchronization signals.
Although there are monitors that already ignore that synchronization waves on the green channel, there are others that don't, and this instructions are of those that don't ignore them.
Step 1: Disclaimer
Do this at your own risk, I'm not responsible for what might happen to your hardware, make sure you do everything with caution, you follow up the instructions correctly and carefully and you always analyze voltage and amperage endings to make sure what you are doing is meeting your hardware specifications.
This mod worked for me, but it doesn't mean that it might end up working the same way for you, i just tested it with my Playstation 2, always making sure that nothing will blow up. But be aware that every hardware has its own specifications and limitations, also every Playstation 2 has its own specifications and limitations, so If you are not an expert, and you don't know what you are doing, don't try to attempt this.
Step 2: Playstation 2 Schematics
If you are interested on how i manage to connect everything, this is the schematic that i used (without the 220uF capacitors, since i found out that my Playstation2 already have them and they are soldered on the RGB lines).
1) 1x LM1881 - Video Sync Separator
2) 1x 74LS04 (aka 7404 Hex Inverter)
3) 2x 0.1uF ceramic capacitor
4) 1x 680k resistor
5) 3x 220uF Electrolytic capacitor (optional)
I bought everything from ebay, those LM1881 and 7404 hex inverter are cheap, i wasted $2 total for 2pcs of each.
Capacitors are easy to find on your old electronics, search for ceramic capacitors that have the number "104" on it.
Step 3: How to Solve the Greenish Problem
Board project: https://circuits.io/circuits/2443955-arcade-ps2-sync-on-green-simple-way-to-remove-greenish
So i added a diode (don't invert the polarization otherwise this will not work) between the Playstation 2 Green and the VGA green pins, this way i can remove the synchronization from green.
But since the diode has an internal resistance, this makes the green wave signal weak, so i added a potentiometer connected to GND and VCC that allows me increase the wave before it gets rid of the sync signal.
In short the potentiometer amplifies the wave, then the diode removes the sync and reduces the wave to it's correct shape.
Step 4: Results
The multimeter allows me to see how much voltage is in the sync removal process and how much do i need, this way you'll always have a reference number that i can work with. For me, in some games like God of War, my multimeter needs to show me 0.490V, for Spider Man i need 0.580V as my reference number, so this will also depend on games.
This is not a perfect solution and you'll still be able to see some remaining green (almost none) in some particular areas, that you can compensate or remove with the potentiometer.
If your game is almost of the same colors, then you just leave the potentiometer as it is, and you keep playing. If you feel like you want to make it work even better, you can use an Arduino to analyze the voltage (like your multimeter is already doing), and increase or decrease the voltage sending an PWM signal like your potentiometer is doing, this way you have an automatic process that increases or decreases the green when you need it.
Step 6: Playstation 2 + GSM
Since this is a Playstation 2, i have to be running Free Mcboot and GSM (Graphics Synthesizer Mode Selector) in order to have VGA output.
If you want to have a clear idea on what i'm talking about, i found out this video on youtube that explains it clearly: