If you value the teak and want to keep the furniture, you have some work ahead of you.
First sand it back to bare wood. If it were me, then, I'd wipe the dust off, and then, I'd wipe it down with acetone, to further remove the natural oil
of the teak from where you want to work. Then, using high quality marine varnish
, built up the coats, 10-12 should see you through. Now for the bad news: every six months, sand it lightly, and add a fresh coat. It will stay looking lovely, but it's a lot of work to maintain.
Oiled teak seems to work well indoors, but outdoors in a humid area, it is a mold
field waiting to happen. The airborne organisms seem to feed on the oxidized oil--at any rate, for me, oiled teak did not work well in the tropics. With varnish
it was fine. Just my experience. I'm only one person, after all.