Clémentine

For English speakers (Yorkshire), we say Cle-men-tyne. I don't know anyone who pronounces the fruit the French way, Cle-mawn-teen, or whatever. Maybe toffs pronounce it like this, like they say I have a meegraine headache instead of a mygrain (migraine) but to me that just sounds pretentious. If you are French, have French ancestors or a genuine reason for using the French pronunciation of the name, then fine. But otherwise, unless you are in France, I would stick with the name as spoken in English speaking countries. Listen to the folk song, if you are unsure of this pronunciation.

I'm surprised that the alternate English pronunciation "Clem-en-tine" (rhymes with "turpentine") hasn't been noted. This is how it's pronounced in the song.

In Britain it's pronounced klem-en-teen, as with the French.

If pronounced the French way (as it is listed as French here) it is kleh-maw(n)-TEEN. Otherwise, I have also heard it kleh-men-TYNE. [noted -ed]

Pronounced "kleh-men-teen".