You say the strata scheme has no registered by-laws. If it was created after 1997 then it would have to have had by-laws lodged with the registration of the strata plan. If it is a pre-1997 scheme, then the model by-laws in the Strata Schemes Management Act apply.
So, presuming it is a pre-1997 scheme, then the model by-law that applies is the one that provides that an owner/resident can’t keep an animal without the written permission of the owners corporation, and that permission can’t be unreasonably refused. From what you say in the third paragraph, it is likely that this is the by-law that applies.
The managing agent is wrong, in that for an owner or resident to keep an animal they must first obtain the written permission of the owners corporation. If they don’t do that, then they are in breach of the by-law. Yes that permission can’t be unreasonably refused, but in some cases an owners corporation may refuse on reasonable grounds. At the extreme, it would generally not be unreasonable to refuse permission to keep a horse.
I am not suggesting that permission should be refused (I have a dog), but the point is that the owner/resident has to ask for that permission and be given it, and the owners corporation may put conditions on the permission. For example, it may say that the animal must be kept within the lot and if it is on the common property it must be under the control of its owner.
In this case, assuming you do have that by-law, and it sounds like you do, the tenants should be told to seek permission, and if permission is given they should be required to keep the cat under control, i.e. off the common property. As they are tenants they should also get the permission of the owner. Of course you could decide to refuse permission, but it would have to be on reasonable grounds, and that can be tricky.
I don’t think cats should be allowed to roam around, they can cause real havoc.