hello i am trying form when i purchase this game to enter into a multiplayer session , but i never ever manage to see any Multiplayer game running .
I have isntalled the GOG galaxy service and such but when i click on the multiplayer session on the game menu I see NO game at all going on. So there should be only two possibilities :

1 ) there are 0 players on GOg server that do multiplayer games

2) there is some setting (TCP ports that need to eb configured on my router)

becouse i think, and hope, that possibility 1) is more unlickely ,i opt for nuber 2) but to solve the issue i need to know what are those ports that i need to configure on my router

CAN any one help ?????

Seconded, same issue for me. Through a steam account, we checked whether the steam version has the same issues, but there, multiplayer games are visible (same laptop). Since crossplay should be possible AFAIK, the same games should be visible from both installations. There are only two differences between the installations:
- the GOG version has all the DLCs installed (besides of the ship designer toolbox), whereas steam version is vanilla
- the GOG version is on 1.42, while steams is on 1.41, although both are up to date

Not sure if this belongs to the issue, but I also had difficulties with the ship designer toolbox DLC. I see the button to register, can enter my email adress and the process is confirmed by the game, but I never receive any email to be confirmed. This ways, I am not able to get the DLC (never getting any redeem code). This ways, I can imagine that the thread opener is right when stating that there needs to be a port configured as it seems like there is no internet connection available at all.

I can see both issues with the GOG version on two different machines in the same network, both running on Windows 10.

I've just installed the latest patch (1.5), which claims to improve on multiplayer. While I can now claim the ship designer DLC (thanks for the fix!), so internet connection is available now, I still can't see anyone else in the multiplayer menu. Also, I don't see how to invite steam friends as mentioned in the change logs as I can't find the button to do so. True: I also don't now where to look for it. I just tried to host a game and searched the GOG galaxy client, but no button anywhere. Any help would be highly appreciated.

GC3 Steam and GC3 GOG are different games regarding multiplayer. You can't play crossover.
They revealed this point in Stardock forum.

GOG's cross platform play currently requires a game to route all multiplayer (Steam and GOG) through GOG's multiplayer service. This isn't something Stardock was prepared to do since Galactic Civilizations III and our other games on Steam have been using Steam's multiplayer successfully for some time. We are very aware having split multiplayer communities is not ideal and hope to address this in the future. I can't share any specific timelines though.

-Adrian from Stardock

Thanks for the reply, at least I know now, that Stardock is aware of the problem and pays attention to it. Hopefully, this doesn't mean that you won't release games on GOG in the future as I prefer it over steam. Just a suggestion: implementing a real LAN mode (I know, I know: no API...) may tweak on the issue a bit as players could play via Hamachi etc and this would also allow real LAN play, which is something that I miss. Just my two cents.