My MIL is a very finicky person and she likes to keep things in order in her own home. But as soon as she comes over to our place, no matter how clean it is, she manages to find fault! How do I tell her, she needs to STOP, without sounding rude or crazy!

Let her do the cleaning if she wants to. Tht way u get to do other things. If she's such a cleanliness buff then no matter how much u scrub around, it will never be clean as per her standard. So best alternative is that you stop and let her do it if she wishes to. Focus ur energy on other things. By doing this u will neither be rude nor crazy....