The most common and most likely answer to this crossword clue is the 6 letter word ETCHES
I've seen this crossword clue in Financial Times
E
T
C
H
E
S
Buddy explains!

The straight part
We know that "Eats out" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case eats out = etches.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 ETCHES mean?
ETCHES verb- To cut into a surface with an acid or other corrosive substance in order to make a pattern. Best known as a technique for creating printing plates, but also used for decoration on metal, and, in modern industry, to make circuit boards.
- To engrave a surface.
- To make a lasting impression.
- To sketch; to delineate.