What on earth is this filmy substance in my tea/mugs ?

[wherehaveallthegoodfolkgone]

Hello all,
Hoping someone can shed light on this frustrating problem. I live in a house where for the past 15 years, whenever I make a cup of tea, there's always an oily looking film floating on the top. No dishwasher. Tried scrubbing and extra rinse of cups. It's so embarrassing when I've got guests. This revolting photo is of my black instant coffee! It's all on the sides of the cups too.
Thanks for any info

We had a water softener fitted to our water system when we had our kitchen done so that the problem was solved throughout the whole house, meaning shower heads, baths, toilets were all easier to keep clean. I was a bit sad because I don't drink tea or coffee but do drink loads of water and hard water is the best water for drinking - all those minerals, I guess. The rest of the household was happy with the lack of scum though.

Definitely not doing the job. I use Brita water jug filters, they work fine