wp-login.php gets downloaded instead of displayed

I didn’t start having this problem until recently. It started when my cookie expired and WordPress forced me to login again. Instead of displaying the login page, it downloaded it (on Chrome). Now I can’t login from anywhere with wp-login.php, not from Chrome, Firefox – on my laptop and on my phone.

I’ve read hundreds of posts all over the web and they’re nonsense in my situation. I’m using Nginx, so there’s no .htaccess to deal with. This only happens with wp-login.php. I’ve cleared everything on all the browsers I use and still no joy.

I copied the wp-login.php to another file, changed all the references within it to the new file name and I can login with it. So it’s not a PHP issue, it’s not an Nginx issue and it probably isn’t a WordPress issue. I’m at a loss as to why this happens.

If anyone else has a clue, please clue me in.