Because court reporters write phonetically, there has to be a way to differentiate all those words that sound the same.  And there is!  All those words can be written a slightly different way.  As they're writing, reporters can differentiate "right" from "rite" and "read" from red."

It's a split-second decision the reporter must make when they write the word, and that requires strict attention to what's going on, what's being said ... in other words, paying close attention, listening for context.

Not all reporters choose to make that differentiation themselves.  They write the phonetics and the conflict appears right in the transcript and also in the realtime display.  In the case of there/their/they're, all those words can be written phonetically as THAIR.  When the reporter "scopes," or proofreads the transcript later on, they resolve the conflicts one by one, choosing which one of those three words is the correct word in each instance.