The most common and most likely answer to this crossword clue is the 6 letter word VERGES
I've seen this crossword clue in The Times
V
E
R
G
E
S
Buddy explains!
The straight part
We know that "Shakespearean constable/couple by the roadside" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case shakespearean constable/couple by the roadside = verges.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.
What does VERGES mean?
VERGES noun- A rod or staff of office, e.g. of a verger.
- An edge or border.
- The phallus.
- An old measure of land: a virgate or yardland.
- A circumference; a circle; a ring.
- The shaft of a column, or a small ornamental shaft.
- The edge of the tiling projecting over the gable of a roof.
- The spindle of a watch balance, especially one with pallets, as in the old vertical escapement.
- To be or come very close; to border; to approach.
- To bend or incline; to tend downward; to slope.