You could always use tell and seek...

my $lpos = tell(DATA);        # remember where I was
while (<DATA>) {
    print "$_";

    if ($_ =~ /another/) {    # if this line has what I want
        seek DATA, $lpos, 0;  # rewind to beginning of line
        print <DATA>;         # and read it again
    }

    $lpos = tell(DATA);       # save offset to the beginning
                              # of the next line
}

__DATA__
This is a line
This is another line
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