I'm heading to Korea later this month and have just spent ages trying to get Kakao T working. It would always get stuck when it would send a SMS verification code to my NZ phone number - I would receive the SMS code within seconds but the app would always say this was wrong.

Support was of zero help but I found a random thread on Reddit which said to add the 0 back into your phone number - so select your country (in my case NZ) and then add the leading 0 back in. I got the SMS still and it worked first time.