I'll have to find the package to see what kind of tropical lily I have, but they resemble star gazers.

and one has some strange offsets on the stem. I really love the blooms on these and would like to successfully propagate.

Here's a link to the stargazers page, it does mentions offsets. Oriental Lily (Lilium 'Star Gazer')

And this is what my plant is doing.

Those are bulbils. After the stems yellow and die, you can harvest the bulbils and plant them. It will take 2 - 3 years for them to grow large enough to bloom.

Just so you know liliums are not tropical plants. It would help if we could see a photo of the blooms. Maybe next year? Bulbils sometimes appear on lilies that don't normally produce them unless the plant is stressed in some way.

Thank you @pardalinum and @DaisyI
I was in a hurry this morning, to not forget to post that. This is the lily species, not sure if the same plant, I have several growing in pots. I have terrible luck with lilies in the ground here. Still haven't found the package they came in. there were 2 varieties in the pack, these are the variety to live and bloom a second year. I think I did put them in the greenhouse last winter.

It does look Star Gazer-ish but lacks the green nectary furrows and the white edge may be a little too narrow. But it does appear to be in the same division as Star Gazer: DIV. VII Oriental lilies.

Don't force the bulbils off; it may take some time before the stem dies back and they loosen up. I give my bulbils a gentle wiggle to determine if they are ready for removal. Then there are those that drop off with a touch and blend in with the soil making them hard to find. This usually happens with some of my Asiatic lilies. Frustrating!

All my orange asiatic lilies have died. They were gorgeous in 2013, and gradually have disappeared, one bloomed in 2019, none this year. They were in the ground and I suspect too much water and or crowding from my John Fanick Phlox choked them out.
The bulbs fell apart on several, I thought maybe, like cloves of garlic, the petals of the bulbs would sprout and I think I potted some but nothing resembling a lily appeared.

When the bulbils are ready to drop, what kind of environment will they require?

I just plant them back by the parent

ok. Thank you. I think the parent is going to get a bigger pot.

If your soil is alkaline, you won't be able to grow lilies in the ground.

Thank you for the explanation. there is limestone under my house.
my soil is alkaline. I managed a few years with the orange asian lilies, and I got like one year on a stargazer years ago.

I can grow spider lilies like crazy, and crinum lilies multiply wildly. So do Amaryllis. (except I think crinums crowded them.)

Besides the bulbils where the leaves meet the stem, you also have stem bulblets near the soil, lower on the stem. Treat them the same.