LDAP is pretty amazing. It lets you set up a bunch of web services that all share a common user database (or rather, directory), so you can have a WordPress installation, a NextCloud, a MediaWiki etc, all of which let you log in with the same user credentials, after you've registered an account on any of them. You don't need to program anything yourself; all this software either supports LDAP out of the box or has well-maintained plugins to offer it.
There's only one problem: LDAP can't help different web services share their login sessions (cookies), so when you log in to one service you're not automatically logged in to any others. You have to log in to each and every service with the same username and password.
So my question: is there a technology that's as widespread as LDAP (in terms of being supported by all kinds of software) which solves the user session sharing problem? Something that's, let's say, supported at least by WordPress, NextCloud, and MediaWiki?