I've seen that too! ie, using STDERR inappropriately to do user communication. But in this case, if that was happening (a print of the question to STDERR), the OP would have seen the question on the screen.

I suspect that in the OP's situation, it could be that unbuffering the output into the STDOUT pipe is a necessary, but not sufficient condition. The tee program also has to use unbuffered output!

I updated my previous post with a link to another thread where I posted a simple program that demonstrates what $|=1 does using only half a dozen print statements.