Here an implementation

Please note how ...

• it avoids slurping the whole (potentially huge) file into RAM
• it's self documenting (well better than one big regex)
• you can now easily add more complicated tests when maintaining

use strict;
use warnings;

my $section;
my $hit;

while (<DATA>) {
    my $start = /^ASDF \{\s*$/;        #(2)
    my $end   = /^\}\s*$/;

    if ($start .. $end) {
        $section .= $_;
        $hit = 1 if /foo_match/;
    }

    if ($end and $hit) {
        print $section;
        $section = $hit = "";           # reset (1)
    }
}


__DATA__
ASDF {
    tmp
    foo_match
    tmp
}
string2 {
    tmp
}
ASDF {
    tmp
    bar_match
    tmp
}
[download]

NB:

• 1) you can also exit instead of resetting
• 2) allowing potential "invisible" whitespace \s* at the end makes it more robust