I know that Möbius strips have applications in real engines. Fanbelts for example are möbius strips because they get worn equally over the whole surface that way.

Perhaps there could be a similar use case for a gear like this,but I have never heard of any.

Usually gears have a prime number of teeth on the cogs involved if possible, and different prime numbers at that.

This ensures that wear is even which helps with the longevity of the parts and also ensures smooth-running as there will be no uneven wear on one part of a cog. Another side effect is to spread oil evenly across all teeth in a reasonably short running time.

I cannot see how a mobius gear would offer any significant benefits over this. Most cogs are already machined to fit each other neatly, and the problem with fan belts (uneven wear) doesn't really apply to gears.

Why would a prime number of teeth be beneficial? Every full rotation, all n teeth should have the same amount of wear.

It's a cool insight that I just learned, but if the two gears share a multiple, then the same teeth will engage with each other over and over leading to uneven wear pattern. If the gears are relatively prime (e.g. 6 and 17) then each tooth will touch each other tooth before repeating the cycle.

Try it out on a piece of paper and see for yourself (that's what I did).

Presumably the desired condition is that gcd(n,m) = 1 (the number of teeth be relatively prime) so that each tooth of one gear meets with each tooth of the second gear, cyclically. If there are variations/defects in teeth, this might help them wear more uniformly.

In an ideal world every tooth on the gear is exactly the same, but in the real world they have slight differences that wear each tooth differently. So by having a prime number of teeth, the teeth of two gears don't mesh with the same teeth on the opposite in a short repeating pattern. The gear teeth wear more evenly this way and imperfections are worn off over time instead of worn in.

I'm not saying that it doesn't exist in perhaps specialized applications but ... I have never ever seen an automotive fan belt that was a möbius strip. Older cars (and other belt driven machinery like drill presses and band saws) use v-belts and newer cars use serpentine belts. Having installed new belts in both my older and newer cars (as well as the drill press and band saw in my shop) I have seen exactly zero that could be called möbius. In fact, I don't recall seeing one at the parts store that was möbius. In fact, now that I think about it a serpentine belt comes in contact with rollers on both faces anyway and would not have to be möbius to wear more evenly.

I don't know where you're based but they were never common in the USA, and where they were common they have since been replaced by modern materials with a clear inside and outside and no twist.