Lookup allotments act 1950. Allows you to at least have a rabbit, or chicken if you really want to. Doesn't matter what's written in the tenancy agreement. The act overrides it and allows them.

Lettings agents like to pretend it doesn't exist and then quietly give up when it's pointed out to them or they check with their legal people.

"Abolition of contractual restrictions on keeping hens and rabbits (1)Notwithstanding any provision to the contrary in any lease or tenancy or in any covenant, contract or undertaking relating to the use to be made of any land, it shall be lawful for the occupier of any land to keep, otherwise than by way of trade or business, hens or rabbits in any place on the land and to erect or place and maintain such buildings or structures on the land as are reasonably necessary for that purpose:Provided that nothing in this subsection shall authorise any hens or rabbits to be kept in such a place or in such a manner as to be prejudicial to health or a nuisance or affect the operation of any enactment. (2)This section shall have effect as from the time when Regulation sixty-two B of the Defence (General) Regulations, 1939, ceases to have effect as respects England and Wales."