The performance on that may not be as bad as you think. I tried benchmarking my read-by-chunks solution against a change-the-input-record-separator-to-space solution. The latter makes the code much simpler, since the only special thing you have to watch for is the newlines. But it was also a bit quicker:

$ perl 1125570a.pl                         
              Rate read_buffer  change_irs
read_buffer 1.15/s          --        -33%
change_irs  1.72/s         50%          --

$ cat 1125570a.pl
#!/usr/bin/env perl
use Modern::Perl;
use Benchmark qw(:all);

# setup long multiline strings with lines ending in 0
my $line1 = join ' ', (map { int(rand()*100) } 1..1000000), 0;
$line1 =~ s/ 0 / 0\n/g;
my $line2 = $line1;

cmpthese( 10, {
    'read_buffer' => \&read_buffer,
    'change_irs'  => \&change_irs,
});

sub read_buffer {
    my $l;                # chunk of a line
    my $tiny_buffer = 1000000;  # buffer size of chunks
    my $leftover = '';    # leftover, possibly partial number at end o
+f buffer
    open my $in, '<', \$line1;
    while ( read $in, $l, $tiny_buffer ) {
        $l = $leftover . $l;
        #    say "        ;$l;";
        $leftover = '';
        if ( $l =~ s/(\d+)$//g ) {
            if ( $1 == 0 ) {
                $l .= '0';
                $leftover = '';
            } else {
                $leftover = $1;
            }
        }
        for (split ' ', $l) {
            if ( $_ == 0 ) {
#                 say 'Reached a zero';
            } else {
#                 say "; $_ ;";   # process a number
            }
        }
    }
}

sub change_irs {
    open my $in, '<', \$line2;
    local $/ = ' ';
    while ( <$in> ) {
#        say "      $_";
        if ( $_ =~ /0\n(\d+)/ ) {
#            say 'Reached a zero';
#            say "; $1 ;";   # process a number
        } elsif ( $_ == 0){
#            say 'Reached a zero';
        } else {
#            say "; $_ ;";   # process a number
        }
    }
}
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The larger the buffer you can use on the read_buffer solution, the faster it should be, I think, but I don't know if it would ever catch up to the $/=' ' solution. Considering how much clearer that one's code is, I think it wins.

EDIT: It also occurs to me that reading the file from disc might make a difference, if the RS=space solution causes more disc reads. I'd think OS buffering would prevent that, but I don't know for sure. You'd want to benchmark that with your actual situation.