`...` is equilvalent to readpipe(qq`...`) (basically readpipe("...")), which is to say it interpolates. As such, you wanted the following:

my $num_subs = `perl -w -0777 -pi -e 'my \$c=s/foo/bar/g;print STDOUT 
+"\$c\\n"' ./foo_test`;
die("Can't execute child: $!\n" if $? == -1'
die("Child killed by signal ".( $? & 0x7F )."\n") if $? & 0x7F;
die("Child exited with error ".( $? >> 8 )."\n") if $? >> 8;

chomp($num_subs);
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The program can be simplified.

my $num_subs = `perl -i -0777wpe'print STDOUT s/foo/bar/g' foo_test`;
die("Can't execute child: $!\n" if $? == -1'
die("Child killed by signal ".( $? & 0x7F )."\n") if $? & 0x7F;
die("Child exited with error ".( $? >> 8 )."\n") if $? >> 8;
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An approach that would make it easier to escape (if we still needed to do so):

use String::ShellQuote qw( shell_quote );

my $cmd = shell_quote("perl", "-i", "-0777wpe", 'print STDOUT s/foo/ba
+r/g', 'foo_test');
my $num_subs = `$cmd`;
die("Can't execute child: $!\n" if $? == -1'
die("Child killed by signal ".( $? & 0x7F )."\n") if $? & 0x7F;
die("Child exited with error ".( $? >> 8 )."\n") if $? >> 8;
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Why are we even invoking a shell?

use IPC::System::Simple qw( capturex );   # Core module

my @cmd = ( 'perl', '-i', '-0777wpe', 'print STDOUT s/foo/bar/g', 'foo
+_test' );
my $num_subs = capturex(@cmd);
die("Child exited with error ".( $? >> 8 )."\n") if $? >> 8;
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Now, one would also quite reasonably argue that one shouldn't execute perl either. However, -i is quite good at avoiding data loss in the event of an error^[1], and recreating this wouldn't be trivial.

Finally, why execute a new perl at all?

my $num_subs = 0;
{
   local @ARGV = "foo_test";
   local $^I = "";
   local $/;
   $num_subs += s/foo/bar/g while <>;
}
[download]

As an added bonus, this last version handles the file not being found better. It also handles multiple files in @ARGV.

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1. This is a recent improvement.