Varnish past it's useful shelf life does that. Just too old. Unfortunately, you will need to remove it, get fresh and start over.

This varnish was just bought a few weeks ago from Sherwin Williams. I couldn't find a date on the can.

And it dries fine on other wood, so doesn't that indicate that it's not a problem with the varnish itself?

Yeah, that does eliminate the varnish. Two things, wipe the surface with a rag wetted with paint thinner. Lightly, do not scrub. Then put a fan blowing on it.

If those do not work, remove it and try a coat of 3 pound cut shellac. If the shellac dries, then try the varnish.

Alas, paint thinner didn't work. I'll mix up some shellac tonight....

Exotic hardwoods like garapa are often very dense and have a high oil content, which can make finishing difficult unless you use something specifically formulated for exotic hardwoods. I'm afraid I don't have a solution to offer (we're primarily a decking company and don't deal very often with this kind of project), but that might be the cause of your troubles.

Well, I experimented a little bit over the weekend with a few samples in my basement.

Sample A: primer + varnish. Primer cured fine, varnish still sticky 24 hours later.

Sample B: primer + shellac. All dry. I put a coat of varnish on last night and it's still sticky this morning.

Sample C: primer, put outside in the sun. Shellac put out in the sun. All dry. I put a coat of varnish on last night and it's still sticky this morning. I've put it outside this morning to see if it cures.

Sample D: primer/outside, varnish/outside. it cured!

So sunlight seems to have had an impact. I put the original piece outside to see if it'll cure this late in the game.

Sample D cured beautifully in today's sunlight with a 2nd coat of varnish, so it looks like we've found the magic formula. I put the handrail piece out and it's dryer than it was, but not as much as Sample D. I'll give it another day and see. Otherwise I'll cut a new piece and start from scratch.