The most common and most likely answer to this crossword clue is the 4 plus 7 letter word YOUR MAJESTY
I've seen this crossword clue in The Daily Telegraph
Y
O
U
R
M
A
J
E
S
T
Y
Buddy explains!
The straight part
We know that "Royal address" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case royal address = your majesty.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.