The most common and most likely answer to this crossword clue is the 8 plus 7 letter word NATIONAL THEATRE
I've seen this crossword clue in Independent
N
A
T
I
O
N
A
L
T
H
E
A
T
R
E
Buddy explains!
The straight part
We know that "South Bank complex" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case south bank complex = national theatre.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.