This thread is resolved. Here is a description of the problem and solution.
Problem:
The client wanted to know how to use a single PO file to apply translations across multiple sites without manually importing it into each site.
Solution:
We recommend manually importing the same PO file into each of the client's sites. Currently, there is no automatic way to synchronize or share PO files across multiple sites. Each site must have the file imported individually to ensure the translations are applied correctly.
Please note that this solution might be outdated or not applicable to your specific case. We highly recommend checking related known issues at https://wpml.org/known-issues/, verifying the version of the permanent fix, and confirming that you have installed the latest versions of themes and plugins. If this does not resolve your issue, please open a new support ticket at WPML support forum.
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This topic is split from https://wpml.org/forums/topic/translate-at-theme-level/