The most common and most likely answer to this crossword clue is the 5 letter word RENDS
I've seen this crossword clue in Financial Times
R
E
N
D
S
Buddy explains!

The straight part
We know that "tears" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case tears = rends.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 RENDS mean?
RENDS verb- To separate into parts with force or sudden violence; to split; to burst
- To part or tear off forcibly; to take away by force; to amputate.
- To be rent or torn; to become parted; to separate; to split.