Run both taps, overwhelmed by high rent (12)
I believe the answer is:
steeplechase
'run' is the definition.
(steeplechase is a kind of run)
'taps overwhelmed by high rent' is the wordplay.
'taps' becomes 'ch' (I've seen this in other clues).
'overwhelmed by' indicates putting letters inside (I've seen this in other clues).
'high' becomes 'steep' ('steep' can be similar in meaning to 'high'**).
'rent' becomes 'lease' (I've seen this before).
'steep'+'lease'='steeplease'
'ch' inserted within 'steeplease' is 'STEEPLECHASE'.
'both' acts as a link.
This may not be correct. It may be part of another bit of the clue.
(Other definitions for steeplechase that I've seen before include "Running event over hurdles and water jumps" , "Race over fences" , "Type of horse race" , "Horse race with jumps and ditches" , "Non-flat horse race" .)