Cross-file repetitions are just segments which are repetitions of the first instance of a particular segment (they have the same content). This segment is placed in the primary file (the file in which a particular segment occurs for the first time). In other words, if you have segments in other files that are repetitions of a particular segment from the primary file, those are called cross-file repetitions.

The analysis of a project takes place in a "top-to-bottom" fashion. When you create a project with multiple files that have the same segment, the first instance of that segment will be marked as Unmatched. Other instances of that segment in other files will be marked as a Repetition with this information: Repeat - matched across files:

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Repetition from another file

If you see this information, it just means that this segment has not been translated yet in any file.

If any instance of that segment in any file is translated then the relevant translation will be inserted in other occurrences if you access them in XTM Workbench, in the "Edit" mode:

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Repetition with the corresponding translation

If a particular cross-file repetition receives a TM match upon project analysis, its first instance will have a ICE/Leveraged status whereas the instances in subsequent files will be assigned Leveraged status and will be displayed in Project Metrics (external matches always take precedence over any kind of internal matching). This is caused by the fact, that the first instance was not marked as a repetition when the file was analyzed:

Although there cannot be a match from the same segment when it has been translated without reanalysis, it is expected that the first occurrence can receive a match from the other segment, which in this case is the child repetition.

Will a segment like this, which is matched to another file that has been removed from the project, be counted as a Fuzzy Match or as a No Match?

That depends which file is removed. If you have two files in a project, one with an unmatched segment and the second one with information: Repeat - matched across files, and you delete the file in which the segment is unmatched, the second file will unfortunately contain an incorrect repetition unless you reanalyze the entire project. In general, when files are added/deleted after project creation, full project reanalysis is required to calculate cross-file repetitions properly.

In XTM Cloud, you can enable the option to hide repeated segments, and this also applies to cross-file repetitions (of course, except for the first instance of the segment). This setting can be enabled in Configuration → Settings → Translation → TM → Repeats:

Once enabled, the option is activated in the "create a new project" form when you can specify the percentage of a particular file at which repetitions occupy, at which the option shall be activated:

This feature in question hides repeated segments and populates the repetitions with the translation of the parent/original segment upon target generation, allowing for great consistency of translation.

Remember that, while repeated segments in XTM Workbench can be hidden, they are automatically populated with the newest translation in subsequent files so they will be translated accordingly in a target file once that file is generated again. Also, the Metrics will still display those repetitions, and the file concerned will show the original number of segments in the Project editor → Workflow section.