What does ~~MULTILATERAL~~ Mean in an imessages text conversation on iPhone?
'Multilateral' generally means multiple participants. If your threads aren't group messages, it's likely referring to the fact that the conversation may be acting like there are multiple participants because you have multiple points of contact for each other (email address, phone number, etc).

I would try following the steps in this article to resolve your issue. If your previous conversations aren't important enough to keep, I would also suggest purging all threads you have for each other on all of your devices before following the steps in the article provided. This just helps with clarity and cutting down on confusion when you begin receiving new messages.

Also, sometimes it helps to review the information you have added in the 'You can be reached for messages at' section of the 'Accounts' preferences panel within the Messages app. Ideally, you should only have your iPhone's phone number and the email address that is associated with your Apple ID listed under this section.