wrote

Hello, our cat seems to eczema (or dandruff, I'm not sure if there's a
difference). Against my better judgement the cat's other parent
started bathing it, and it was a little while after this that the
eczema started.

suspicious of the whole cat-bathing concept. The vet prescribed some
tea-tree-oil shampoo, so now the cat is being bathed with that, which
cures the problem for a short time, but it soon comes back, needing
more bathing.

Groan. First, please stop bathing the cat. Though there can be reasons to
have to do it (heavy flea country for example like where I am so a true flea
dip may be required), it shouldnt be done gratuitously. Reason is the cat's
skin doesnt produce the right oils enough to withstand much bathing.
Result? Exactly what you have going on.

Second, you may want to reconsider that vet. Selling you tea tree oil
shampoo?

Third, treatment is real simple. Before going any expensive route, this is
almost sure to work and will *not* harm the cat. Slightly more expensive
version is omega 3 oils which one here recommended. Less expensive version,
start adding fats (oils) to your cat's diet. It doesnt have to be much. 1
TS of olive oil per meal is what I mostly use. Thats probably more than
needed but harmless.

Adding to that the best is to vary the oil/fat a little bit. Others use the
oil from an oil packed tuna can (which I do too but only now and again). I
vary this with other things like a little dab of duck fat (I cook duck about
every 6 weeks so save the fat for cooking and the cat and dog get a little
as a treat). Those little fat globs inside a chicken? I peel out a little
and nip it up then add just a pinkynail worth for the next day or so (holds
fresh in the fridge about 3 days). Bacon fat works too. The bacon fat one
they really like. Don will have bacon now and again and once the pan is
cooled, we pour off the fat then let them lick the pan g.

I have a high shed cat and dog but have no furball problems or skin
problems. I've had many other cats over time and some of them were slightly
sensitive to some oils. Sesame oil for example isnt recommended (they dont
like it much and it's prone to making them have an unhappy tummy). Corn oil
was a problem for one of them (gave'em the runs, probably corn sensitive
cat). I've heard of a few that are slightly sensitive to olive oil but this
one is easy to have handy and rare to react badly to (I've never had one
react badly to it). Rendered meat fat is never a problem unless it's a
salty meat (caution, pets arent very salt tolerant so watch the bacon fat
and use that just now and again unless you use sodium free bacon like we
do).

Cheapest method: Bake a plain chicken and pour the fat off into a jar.
Keep the jar in the fridge (chicken fat will go rancid if not kept that way
unlike bacon fat). Scrape off a little pinky-nail sized portion and add it
ontop of the cat's food (dry or wet). Need not be melted, they like it cold
just fine and will lick it up happily. Vary this with other saved unsalted
meat fats such as from a fresh (unbrined! Unsmoked!) pork shoulder. Work
into your cooking, the occasional meal that you don't spice or salt until
after cooking and save the fats. Free pet food oil g.

Hope this helps you or others!