A Zip archive that contains files with absolute paths is strictly non-compliant with the zip specification. Below is taken from here

       4.4.17.1 The name of the file, with optional relative path.
       The path stored MUST not contain a drive or
       device letter, or a leading slash.  All slashes
       MUST be forward slashes '/' as opposed to
       backwards slashes '\' for compatibility with Amiga
       and UNIX file systems etc.  If input came from standard
       input, there is no file name field.
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I assume that Windows Explorer is using the strict interpretation of the zip spec.

The Archive::Zip docs do cover this in the addFile section

    NOTE that you should not (generally) use absolute path
    names in zip member names, as this will cause problems
    with some zip tools as well as introduce a security
    hole and make the zip harder to use.
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As mentioned already in Re: Archive::Zip problem with Windows Explorer Windows Explorer can display both the path & filename if it uses a relative path.

Something like this (untested) should allow you to store your files using a relative path.

  foreach my $href ( @files_to_get ) {
    my $file = $href->{"files"};
    print( "  adding $file\n" );
    my $name = $file;
    $name =~ s/^[A-Z]://; # remove drive
    $name =~ s#^[/\\]+##; # remove leading path delimiter
    $zip->addFile( $file, $name );
  }
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