The Binding of Isaac: Rebirth gets online co-op in the latest update with Repentance+

Available in Beta for The Binding of Isaac: Rebirth, is the new The Binding of Isaac: Repentance+ update that brings online co-op.

You need to own all the DLC for it, and they're classing it as a Beta for now while they refine it and squash remaining bugs. Additionally, as far as I know, while the base game has Linux support you need Proton to run this because the Repentance DLC is only supported on Windows.

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Online play supports all game modes, all achievements that you get in solo can be earned while playing online but overall there's a few differences from solo play as they detailed in the release notes:

Online play has a few differences from solo play. Defeated bosses will drop an item for each player in the match. If a match is joined in progress, the new player will be granted several items to catch them up based on how far along the run is. Players can spend coins at the revive machine in the starting room of each floor to bring back dead players. Be warned, however, that the cost goes up each time the machine is used. Defeating a boss at the end of the stage will also revive any dead players. There might be some other special things to discover online as well!

Something to note is that mods are not supported to play online, you need to disable any mods.

Did they change anything for solo players? Yes, lots. There's various item improvements and reworks, plus many general quality of life fixes and bug fixes. It's a big update overall.

Doesn't seem like GOG have the update yet, so stick to Steam if you want all the goodies.