The most common and most likely answer to this crossword clue is the 8 plus 4 letter word PRINCESS ANNE
I've seen this crossword clue in The Guardian
P
R
I
N
C
E
S
S
A
N
N
E
More clues leading to the result PRINCESS ANNE
Buddy explains!

The straight part
We know that "royal issue" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case royal issue = princess anne.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.