After googling quite a bit, and finding only questions about this
error in CURLOPT_VERBOSE logs, I believe I may understand what this
message means in general.

Just a theory, mind you.

The clue is in trying to use CURLOPT_FOLLOWLOCATION when open_basedir
restriction is in effect.

It explicitly prints an error that you can't do that.

When I try to use CURLOPT_HEADER (so I can manually follow the
Location: headers) it get "Failed to open/read local data from
file/application" in the logs for every request.

Since I don't get that on my laptop without open_basedir, and do get
it on the shared host with open_basedir, I'm guessing CURLOPT_HEADER
needs file access, but is blocked by open_basedir.

It's also true of the COOKIE settings, I think.

Just a theory...

I'd sure like a better answer if it's out there and I missed it.

-- 
brain cancer update:
http://richardlynch.blogspot.com/search/label/brain%20tumor
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