Epitaph soon started to produce tears (4)
I believe the answer is:
rips
'tears' is the definition.
(I know this)
'epitaph soon started' is the wordplay.
'epitaph' becomes 'RIP' (rest in peace).
'started' indicates taking the first letters.
The first letter of 'soon' is 's'.
'rip'+'s'='RIPS'
'to produce' is the link.
(Other definitions for rips that I've seen before include "cuts" , "Tears violently" .)