Do management consulting firms allow individuals to have their own side business?

It depends on what the side gig is. Generally, they do not want you focused on other jobs given you won’t have time to do both and consulting is very busy.

However, it can vary by firm and what the side gig is of course. Sometimes they will allow this as long as there is not a conflict of interest.

For example, if you are a money manager and your client at the consulting firm is “Company XYZ” and you tell your side business clients, invest/don’t invest in “Company XYZ” due to insight you have to fluctuations in the company, then that would be considered a conflict of interest (insider trading).