I am using the pre-reformulated Mobil Vactra. It and many others are available via mail order. I bought mine for $11 US from Enco with free shipping (inside the US) a few years back.
Barring that, if you want a home-brew, take some of that same motor or hydraulic oil and add STP or Lucas oil additive. It will increase the clinginess, much like the tacifiers that are added to purchased way oil products.
Here is a photo of pure Lucas oil additive on the end gears on my lathe. I just drip a drop or two on and it clings nicely while lubing better than grease which squeezes out. Notice the nice web it formed between spinning gears: