Pronunciation of Korea and Career
Solution 1:
As a speaker of Southern Standard British English (RP), these two words are homophones for me. They are both pronounced /kə'rɪə/.
However, SSBE is non-rhotic - we only pronounce /r/ before a vowel sound. For speakers of rhotic Englishes, for example General American, or some regional varieties of British English (e.g. speakers from Scotland or the south west), there will be an /r/ at the end of career, but not Korea.
In addition, in some varieties of English there is likely to be a full /ɒ/ sound (like the vowel in hot) in the first syllable of Korea, whilst there will still be a schwa, /ə/, in the first of career. This is likely to be true of speakers from Yorkshire, who are likely to pronounce Korea as /kɒ'rɪə/. These speakers also have a full /ɒ/ in word beginning with unstressed con- such as concern: /kɒn'sɜ:n/.
In short, for many non-rhotic speakers career and Korea are homophones, they are pronounced the same. But for rhotic speakers and some non-rhotic speakers the pronunciations might be noticeably different. For these speakers career and Korea will be minimal pairs, where some feature of the pronunciation is sufficient for other speakers to be able to tell the two words apart.