Tomcat / mod_rewrite Conflicts with extensionless requests

I have been having too many fruitless conversations with support representatives lately, so I figured maybe I'd have a better chance getting an answer here. This is a complicated problem though; I didn't just set it at 500 points for no good reason (which I sometimes do).

One of my websites is hosted by GoDaddy.com, using their "Deluxe Linux Plan":
https://www.godaddy.com/gdshop/hosting/shared.asp?app%5Fhdr=&ci=5652#tabs

I created my whole website in PHP and had it running just fine. I even used some mod_rewrite rules so that certain files could be called without a file extension. For instance, this one:

    RewriteRule ^search$ search.php [NC,L]

You probably all know how that works already, if someone asks for mydomain.com/search it will load up search.php, even though they didn't ask for that.

So everything worked. In fact, everything worked *perfectly*. But then I decided that I wanted to add some special kinds of functionality to my site--in particular, functionality that can be achieved better with Java than with PHP. So I went to the hosting control panel, and clicked the button to "enable support for Java."

This takes a little while to process, and effectively turns on Apache Tomcat. I know very little about Tomcat, just that it's needed for Java in many cases. Ever since I did that, my site doesn't really work anymore.

Any RewriteRules in .htaccess are completely ignored for extensionless requests, and I based a good deal of my site on such requests. So now, if I try to access mydomain.com/search, I get a 404 error instead of seeing the search.php as I should.

I found the article below on the GoDaddy Help Center. It has a long list of file extensions that are handled by Apache, while "files not appearing in this list are processed by Tomcat" and therefore "bypass [the .htaccess] settings." And since my "extensionless" requests obviously are not on this list, it is clear that any corresponding RewriteRules are being bypassed completely.
http://help.godaddy.com/article.php?article_id=1133

WHAT I WOULD LIKE TO KNOW:
Is there some fix for this? Some way to tell Apache/Tomcat/Whatever that it should process the htaccess settings for "extensionless" files? And more importantly, is there some concise set of instructions I can tell the support reps to do in order to resolve this issue?

Thanks very much in advance. You don't want to know some of the answers that the support reps gave me. I really appreciate any help you guys can offer.