I would like to achieve the following: Run a command and capture its output. While waiting for output, run a subroutine every 10 seconds.

Sounds like exactly the kind of job IPC::Run can do:

use warnings;
use strict;
use IPC::Run qw/ harness timer /;

my @cmd = ( 'perl','-le','sleep 60; print "Done!"' );
my $h = harness \@cmd, \undef, \my $out, \my $err,
    ( my $t = timer 10 );

my $startt = time;
$h->start;
printf "%2d started\n", time-$startt;
while ( $h->pumpable ) {
    $t->start;
    $h->pump;
    printf "%2d pumped\n", time-$startt;
}
$h->finish;
printf "%2d finished\n", time-$startt;

print "stdout: <<$out>>\nstderr: <<$err>>\n";
[download]

You can also capture the "live" output of the command, The $out and $err variables should be updated as data from the external program comes in.

Update: See IPC::Run::Timer for details on the timings (on my system the delay in the above ends up being about 11 seconds).