Is it possible to disable the anonymous posting/replying feature?  I'd like to have some control over who can post and reply on my forum. Is this possible?

Yes it is possible, for that it is enough to activate the privatization of the forum in configuration.
Create at least one moderator, so users log in with this password.
Many Flatboard users use this solution.

Flatboard is based around Anon posting.  I think GooBBS, at one point, had users login.  

I guess you can add an option to "Disable Anonymous Posting".  Don't think this should be a script to log in by default.

 
I was just hoping that there was a choice to allow or not allow anon posting.  

This is confusing to me.  I enable private forum but no one can even see any postings without logging in.  When I try to create a password to register with I get wrong password error message.  How do new users register for the forum?  Maybe I am misunderstanding what this script is for.  I thought it was a forum script where users can register with a password and post/reply to others similar to other forums.  Even if they choose to be anonymous they can still read the forum posts (even in private mode) to decide whether they want to register or not.  Sorry for my confusion but I was hoping for a simple flat file forum script instead of using a database driven one.  

Okay I figured out that new users have to create a username@password in able to post in private mode as well as in normal mode.  New users cannot see the discussions already in the system however when in private mode.  Not sure why anyone would even want to join a forum if they couldn't see the discussions first before commenting or registering.  For now I'll use normal mode to learn more about your script to see if it is what I am looking for. Sorry for my continuous posting but I am trying to figure it all out.

I will study the question to see how to do it.
I put aside your suggestion.