Have you tried cleaning the stains with a magic eraser? Not sure if that will work but it would be worth a try. I know I wouldn't be able to live if I didn't have those things in my kitchen.

Another potential cause of the stains could be that the stainless is touching a different type of medal. Say for instance if the handles of your refrigerator are aluminum, the cheaper metal can react with the more expensive metal and cause rust stains. I doubt this is the problem but it's just a thought. Most likely it's what you said about the tea.

The following assumes that the stainless steel in question is not coated.

DuPont 5717S, available at automotive parts stores that deal with body shops, contains phosphoric acid that works well for cleaning steel. Be sure the room is well ventilated. Protect the floors. Wear gloves and eye protection, and dilute per directions into a non-metallic container. Put some of the diluted 5717S on a cloth pad. Wipe down the stainless steel and rinse with fresh water using another wet cloth and dry. Rinse everything touched by the acid.

If it still exists, PPG Metalprep is equivalent to 5717S.

This acid, as diluted, is not as aggressive as sulfuric, hydrochloric, cresylic, or hydrofluoric acids, but will be irritating to painful if left on the skin for more than a few tens of seconds. I would put its hazard at the same level as that of Clorox or ammonia (not both mixed).

Although I normally use it for car body repair, it proved handy when nothing else would remove some factory fingerprints on my Wolf hood extension.

kas

Two things you can try (make sure fridge is real SS before using the BKF). Try some Barkeeper's Friend (the powdered version seems to pack more oomph) on a sponge/blue scrub sponge (not the green, which is too abrasive), and gently "scrub" the stain/streaks (going with the grain in the metal). Usually that will do the trick. Alternatively, try some Goo Gone (or Goof Off, nastier smell though) on a piece of paper towel or a rag. You can follow up with some BKF to remove any oily residue from the Goo Gone/Goof Off.

Might take a few rounds of cleaning, but should take care of the streaks. A dear friend had recently renovated her kitchen, and shortly after completion, her DH or one of the kids got mustard on her brand new SS DW. Even after cleaning it up, there were streaks on the SS. I cleaned them off for her (pretty sure I just used BKF), following up with my alcohol-water-lavender oil spray. DW was good as new!

Tip: skip the so-called SS cleaners/polishes. If you've already used them, remove it all (residue) using BKF and blue scrub sponge, "rinsing" afterward with a wet paper towel, cloth, rag, etc. Again, might take a few passes to get back to "bare" SS. Then, use the alcohol spray (in a spray bottle, mix ~30%-50% rubbing alcohol to ~70%-50% water, plus enough drops of essential lavender, or your choice oil to scent the mixture as desired), and a microfiber cloth for cleaning. I use this mixture on all of my SS, and on my granite. Works great!

HTH!

A belated, but sincere THANK YOU for the suggestions. The BKF, which I have and use frequently....never would have thought it gentle enough for stainless---nor a blue scrubbie. And yes, I've used Stainless Cleaners so looks like I have some cleaning to do.
Interesting that you suggest microfiber---I hate it for my granite because it feels like it just smears stuff around. I've gone back to using out-of-service terry towels. And, maybe the smearing is coming from the specialty cleaners---perhaps that won't be an issue with your alcohol mix you suggest! I'll give it a try. And there's always Kaseki's more aggressive solution to try if round one doesn't work. Thanks, all!

Samsung SS fridge with water marks under water dispenser. Bar Keepers Friend saved me.