Definition of rein noun from the Oxford Advanced Learner's Dictionary

rein

noun
reɪn
 
; reɪn
 
 
1 [countable, usually plural] a long narrow leather band that is attached to a metal bar in a horse's mouth (= a bit ) and is held by the rider in order to control the horseShe pulled gently on the reins.2 reins [plural] (British English) strips of leather, etc. worn by a small child and held by an adult in order to stop the child from walking off and getting lost3 the reins [plural] the state of being in control or the leader of somethingIt was time to hand over the reins of power (= to give control to somebody else).The vice-president was forced to take up the reins of office.
Idioms

give/allow somebody/something free/full rein

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give/allow free/full rein to something

to give somebody complete freedom of action; to allow a feeling to be expressed freely
The designer was given free rein.The script allows full rein to her larger-than-life acting style.When I paint I just give my imagination free rein.
more at keep a tight rein on somebody/something at tight