You're right. You'd have to doing something like:

my $t1 = async{ 1 while <$out> };
my $t2 = async{ 1 while <$err> };
$_->join for $t1, $t2;
...
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However, if your redirecting stdout to stderr, is there any point in reading stdout? And if you're going to discard the output without looking at it, why not just dump the whole lot to nul and have done with it?