The problem is that the  my $handle variable is private to the callback sub, and is not referenced anywhere else. So when that sub returns (immediately after push_writing your "Hello" message), it will be destroyed, and with it, your connection.

A solution is including a reference to it in one of AE::Handle's callbacks, e.g.:

        on_error => sub {
            my ($hdl, $fatal, $msg) = @_;
            AE::log error => $msg;
            $hdl->destroy();
            undef $handle; # <------
        },
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This way, the callback sub closes over the variable, so its reference count will be nonzero when the sub returns and thus your connection will stay alive.