Is there a word or words to describe a workplace where employees end up managing managers?

Solution 1:

This dynamic is called Managing Up.

Bad bosses are the stuff of legend. And too many managers are overextended, overwhelmed, or downright incompetent — a topic that HBR has covered extensively over the years. Even if your boss has some serious shortcomings, it’s in your best interest, and it’s your responsibility, to make the relationship work.

HBR recently ran a special series on managing up, asking experts to provide their best practical advice for navigating this important dynamic.

(From What Everyone Should Know About Managing Up by Dana Rousmaniere, Harvard Business Review, January 23, 2015)

Employees are having to take on managerial tasks not in their job role because it's a managing-up workplace.

Solution 2:

One word which might resonate is dysfunctional:

Not operating normally or properly.

Government stonewalling and a dysfunctional justice system also jeopardized the case.
It's not a threat to write a book about a dysfunctional intelligence organization.

— Lexico

That particular dictionary is happy to describe physical objects which don't work as "dysfunctional", but I'd prefer to reserve it for systems which don't work well (like the way a company should be managed).