A Former Manager Is Giving References Under My Name

Location: Washington. I used to manage a small customer service team. I left that job in March and have no role with the company now. A former coworker called me yesterday to thank me for the glowing reference I supposedly gave her to a prospective employer. I was confused because nobody had contacted me. She forwarded an email from the hiring company quoting a reference from "team manager" followed by my full name. The comments mention projects I did supervise, but they also describe an incident that happened months after I left. The company email address in the chain belongs to my former supervisor. It looks as though she answered the reference questions and signed my name at the bottom. My coworker is upset because part of the reference is inaccurate, and she thinks it might affect the offer. I feel bad for her and I am also bothered that my name was used without asking.

I emailed the supervisor asking whether she had sent the reference and requesting a copy of what was submitted. She replied, "We had to reply quickly and you knew her work best," but did not actually say whether she signed my name. I have saved the forwarded email and her response. I have not contacted the prospective employer because I don't know whether that would create more trouble for my coworker. I'm not looking to make a public accusation, and I dont want to complicate someone else's hiring process, but I want to stop this from happening again. Is there a practical way to put the company on notice that it cannot use my name for references? Does it matter legally that the statement was positive overall if some details were false and I didn't make it?