grammar - How should I avoid this reflexive pronoun, or is it okay?

I typed a sentence in Microsoft Word as:

• Each mobile operating system implements encryption in their own way.

It was part of a list of bullet points. I got dinged for their own being grammatically incorrect, and the error was

Reflexive Pronoun Use (consider revising)

My question: What is a reflexive pronoun? Also, is there a better way to word this, or is it fine as is? I know software won't have all the nuances of the English language correct, but as I don't know who will be viewing this document in the future, I'd like to have all my Pronouns in a row.

Answer

Rule One: Never accept what Microsoft's grammar checker says as gospel.

In this case, it has mis-identified their as a reflexive pronoun. That name is generally reserved for pronouns like themselves.

However, it has correctly identified that their does not match anything else in the sentence, which is all singular. While you can use their as a singular sexless pronoun [according to many], in this case "Each mobile system" does not need a sexless pronoun, and its would be correct.

• Each mobile operating system implements encryption in its own way.