This is a variation that I tried not to muddle but I'm sure that I have the last part right. I thought only semi-colons (is, however) come before a conjunctive adverb.
Experts agree that the South American apple, a red fruit grown in the summertime, inspired the North American apple; exactly how the South American apple came to North America is, however, unknown.
Hi, Clueless—I agree with Ahmed's observations above. There are actually four possible places you could put "however" in the second half of the sentence: initial, medial (two possibilities), and final position. Only in initial position can you use the semicolon before it:
(1) Experts agree that the South American apple, a red fruit grown in the summertime, inspired the North American apple; however, exactly how the South American apple came to North America is unknown.
(2) Experts agree that the South American apple, a red fruit grown in the summertime, inspired the North American apple; exactly how the South American apple came to North America, however, is unknown.
(3) Experts agree that the South American apple, a red fruit grown in the summertime, inspired the North American apple; exactly how the South American apple came to North America is, however, unknown.
(4) Experts agree that the South American apple, a red fruit grown in the summertime, inspired the North American apple; exactly how the South American apple came to North America is unknown, however.
It is also perfectly fine to punctuate all four sentences above, including (1), with a period instead of a semicolon. In variations (2), (3), and (4), "however" interrupts the sentence, as Ahmed explained, and works as "though" would. Indeed, in variations (2) and (4), "however" could replaced with "though":
(2a) Experts agree that the South American apple, a red fruit grown in the summertime, inspired the North American apple; exactly how the South American apple came to North America, though, is unknown.
(4a) Experts agree that the South American apple, a red fruit grown in the summertime, inspired the North American apple; exactly how the South American apple came to North America is unknown, though.
Now, I'm not convinced that your semicolon is desirable in any of these sentences. I'd advise you always to err on the side of caution with semicolon usage. To combine the two clauses into one, it would be more desirable to introduce the second with "though" and thereby render it subordinate:
(5) Experts agree that the South American apple, a red fruit grown in the summertime, inspired the North American apple, though exactly how the South American apple came to North America is unknown.
(6) Experts agree that the South American apple, a red fruit grown in the summertime, inspired the North American apple, though it is unknown exactly how the South American apple came to North America.
Notice that "though" is (and must be) preceded by a comma in (5) and (6).