my $analysis_file = shift;

open INFILE_1, '<', $analysis_file
    or die "Can't open '$analysis_file': $!\n";

while(<INFILE_1>) {
    my $ALL_FT_FILE = $_; # or some part of $_ after munging

    open INFILE_2, '<', $ALL_FT_FILE
        or die "Can't open '$ALL_FT_FILE': $!\n";

    while(<INFILE_2>) {
        ...  # is the outer $_ clobbered here?
    }
}