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use strict;

my @filenames = qw/one.txt two.txt three.txt/;
...
    } ## end else [ if ( join '', @lines )
} ## end sub read_lines_parallel

first line file 1
|line 1 file 2
|first line file 3
...
--------------------
||fourth line file 3
--------------------

while (1) {
    my $lines_ref = read_lines_parallel(@filehandles);
    last unless $lines_ref;
    print join '|', @$lines_ref;
    print '-' x 20, "\n";
}

while (my $lines_ref = read_lines_parallel(@filehandles)) {
    # Do something with $lines_ref or @$lines_ref here
    print join '|', @$lines_ref;
    print '-' x 20, "\n";
}