Might they produce tears or tiers? (4)
I believe the answer is:
rows
This clue is a double definition.
'might they produce tears' is the first definition.
(arguments might end in tears)
'tiers?' is the second definition.
(tier is a kind of row)
'or' acts as a link.
(Other definitions for rows that I've seen before include "Arrangements side by side in lines" , "Columns" , "Arguments" , "emotional exchanges" , "Quarrels; ranks" .)