In CGEL, while discussing the purpose adjuncts "so/in order" + that-clause (p. 727), it is stated that "that" is omissible after "so", but not after "in order".
Under result adjuncts, there is "so/with the result" + that-clause (p. 732). There are examples with "so" without the "that", but there is just one example with "with the result", and I can't decide if it sounds correct without the "that":
They had gambled away all their money, [so /with the result that they didn't even have the fare to get home]
Contrary to the purpose adjuncts, it is never made explicit whether or not "that" is omissible after "with the result".
Thanks in advance.