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It is not. Please read the first sentence of the license: https://gitlab.com/gitlab-org/gitlab-ee/blob/master/LICENSE#...

Here is how open source is officially defined: https://opensource.org/osd-annotated

Edit:

I did not downvote you. Your point is valid because it reflects a common view in the community. Asking questions should not be punished. Please consider this before downvoting the comment above.

What's stopping someone from simply cloning that directory and putting it on their servers like with the Gitlab community version? How do they enforce the subscription access?

It's simply illegal to do that. A private person may do something like that (even it's illegal), but I can't think of a serious company which uses priated software.

GitLab EE is closed source, but publicly viewable and the license allows for modifications, but you need to be a customer

https://gitlab.com/gitlab-org/gitlab-ee/blob/master/LICENSE "have a valid GitLab Enterprise Edition subscription for the correct number of user seats"

The source is publicly viewable but neither Free Software or Open Source.