Is there a general rule for whether adjuncts who are regular faculty in other departments can supervise grad students?
In our department (Ohio, US) the rule is that to supervise a graduate student, you need to go through a certain "credentialing process" (should have a certain number of publications in the area, hold a degree in one of several possible fields related to mathematics, etc.) so if somebody from other departments wants to supervise our students, all he formally needs to do is to submit an application, have his credentials verified, and go ahead. It is not something that happens often but it does sometimes and right now I'm trying to figure out how it works with co-advising from a different country. So, it is nothing unheard of but the exact rules and procedures may vary widely, which means that your best bet is to ask the department chair or secretary. The department websites usually do not display such things. One should read thick regulation handbooks for faculty to find them out and there is no reason why you should dig into any of those yourself instead of asking people whose duty is to know all such stuff or exactly where to look for it.