[Sony Xperia 10 III] combinedfleet:
This appears to be a a software or firmware issue, rather than faulty hardware. I can consistently and reliably "fix" this (until the next time I lock/unlock the screen) by launching an app from the whitelist in

com.sonymobile.displaybooster

while the screen's brightness is set to maximum and either having your color gamut set to "Original mode" or having the "Video image optimization" option set to enabled.

This is a bit of a dense statement, so let me rephrase.

Each time you turn on the screen, you'll get the green tint. To get rid of it, you should go into the display color gamut settings and either choose the "Original mode" or enable "Video image optimization" (this only has to be done once).

With this config in place, you can fix the green tint by setting your brightness to maximum and launching an app from the list, and that's it - after this trick you'll get perfect display quality even at the lowest brightness setting.

For reference, here's the whitelist:

<booster_setting>
<target_package>
<package_name>com.sonyericsson.album
</package_name>
<hashed_signature>FF61EDA8F583DF412C5DFF08D5575A9976F25CE63B9E06CE2BC48FC44968FBCF
</hashed_signature>
</target_package>
<target_package>
<package_name>com.sony.tvsideview.phone
</package_name>
<hashed_signature>4276919680F4ABA6D41924031C20AC2624FEEA5E8EC68A36529B2ACB664D911E
</hashed_signature>
</target_package>
<target_package>
<package_name>com.google.android.youtube
</package_name>
<hashed_signature>3D7A1223019AA39D9EA0E3436AB7C0896BFB4FB679F4DE5FE7C23F326C8F994A
</hashed_signature>
</target_package>
<target_package>
<package_name>com.google.android.videos
</package_name>
<hashed_signature>3D7A1223019AA39D9EA0E3436AB7C0896BFB4FB679F4DE5FE7C23F326C8F994A
</hashed_signature>
</target_package>
<target_package>
<package_name>com.google.android.apps.photos
</package_name>
<hashed_signature>3D7A1223019AA39D9EA0E3436AB7C0896BFB4FB679F4DE5FE7C23F326C8F994A
</hashed_signature>
</target_package>
<target_package>
<package_name>com.amazon.avod.thirdpartyclient
</package_name>
<hashed_signature>2F19ADEB284EB36F7F07786152B9A1D14B21653203AD0B04EBBF9C73AB6D7625
</hashed_signature>
</target_package>
<target_package>
<package_name>com.netflix.mediaclient
</package_name>
<hashed_signature>363863596EA99241EB71B1A985553AA604DE3EA3C5F0C546742390E682164E6B
</hashed_signature>
</target_package>
<target_package>
<package_name>com.netflix.mediaclient
</package_name>
<hashed_signature>DFABBEB86AD8E315EA19BB022993166C9A701F7879C0B8D98B4D94F988B0A5BE
</hashed_signature>
</target_package>
<target_package>
<package_name>com.sonymobile.colorgamut
</package_name>
<hashed_signature>FF61EDA8F583DF412C5DFF08D5575A9976F25CE63B9E06CE2BC48FC44968FBCF
</hashed_signature>
</target_package>
<target_package>
<package_name>com.playstation.video
</package_name>
<hashed_signature>69D3595A7EFEC5818E299C6C8D71B06F701A07A49ED77401F022E5F738723FE4
</hashed_signature>
</target_package>
<target_package>
<package_name>com.dazn
</package_name>
<hashed_signature>B435100EFE3E10B4B9F990C7ABF2A12BD83D024B405CB308350AAB8950D57732
</hashed_signature>
</target_package>
<target_package>
<package_name>com.sonyericsson.android.camera
</package_name>
<hashed_signature>BA96BD54B5E71823644C826309AB4B2AAF2543021779E40D46F5A151BDCE4F62
</hashed_signature>
</target_package>
<target_package>
<package_name>jp.co.nttdocomo.dtv
</package_name>
<hashed_signature>6CE0AAD7C09B4491B9E581AF45674F8C73C79C947B70092ED29178876778B38A
</hashed_signature>
</target_package>
<target_package>
<package_name>com.sonymobile.cinemapro
</package_name>
<hashed_signature>BA96BD54B5E71823644C826309AB4B2AAF2543021779E40D46F5A151BDCE4F62
</hashed_signature>
</target_package>
<target_package>
<package_name>com.sonymobile.cinemapro
</package_name>
<hashed_signature>BC01A8CD9E5D87854F6DC4C84AED49EDC34AC196C00B89623CEA6CCBBDEA627B
</hashed_signature>
</target_package>
<target_package>
<package_name>com.sonyericsson.album
</package_name>
<hashed_signature>6339375AC295CB0CD22811B97ACCD40104BD4A0185D4DD2289B81860C15D623C
</hashed_signature>
</target_package>
<target_package>
<package_name>com.sonyericsson.android.camera
</package_name>
<hashed_signature>BC01A8CD9E5D87854F6DC4C84AED49EDC34AC196C00B89623CEA6CCBBDEA627B
</hashed_signature>
</target_package>
<target_package>
<package_name>com.sonymobile.extmonitorapp
</package_name>
<hashed_signature>BA96BD54B5E71823644C826309AB4B2AAF2543021779E40D46F5A151BDCE4F62
</hashed_signature>
</target_package>
<target_package>
<package_name>com.sonymobile.extmonitorapp
</package_name>
<hashed_signature>BC01A8CD9E5D87854F6DC4C84AED49EDC34AC196C00B89623CEA6CCBBDEA627B
</hashed_signature>
</target_package>
</booster_setting>

In my experience, com.sonyericsson.album works best because it exposes the activity

com.sonyericsson.album.video.player.PlayerActivity which is convenient because it plays a video and then exits immediately. So I've implemented a hacky workaround in Termux which basically boils down to cranking the brightness up to maximum, opening a very short video which consists of a single black frame using said activity and then setting the brightness back to normal. Termux then allows you to add such a script to your home screen, which means now you just have to tap the icon after unlocking your phone to fix the display.

Here's the script:

#!/data/data/com.termux/files/usr/bin/bash
termux-brightness 255 && sleep 0.5s && am start --activity-exclude-from-recents -a android.intent.action.VIEW -d /storage/emulated/0/Download/vid.mp4 com.sonyericsson.album/.video.player.PlayerActivity && termux-brightness auto

(Replace the path to the video file with a path to a suitable video file, I'll attach mine to this post for convenience)

Hopefully a real fix is delivered by Sony in the next firmware update.