We introduce a new algorithm for finding stable matchings in multi-sided matching markets. Our setting is motivated by a PhD market of students, advisors, and co-advisors, and can be generalized to supply chain networks viewed as $n$-sided markets. In the three-sided PhD market, students primarily care about advisors and then about co-advisors (consistent preferences), while advisors and co-advisors have preferences over students only (hence they are cooperative). A student must be matched to one advisor and one co-advisor, or not at all. In contrast to previous work, advisor-student and student-co-advisor pairs may not be mutually acceptable (e.g., a student may not want to work with an advisor or co-advisor and vice versa). We show that three-sided stable matchings always exist, and present an algorithm that, in time quadratic in the market size (up to log factors), finds a three-sided stable matching using any two-sided stable matching algorithm as matching engine. We illustrate the challenges that arise when not all advisor-co-advisor pairs are compatible. We then generalize our algorithm to $n$-sided markets with quotas and show how they can model supply chain networks. Finally, we show how our algorithm outperforms the baseline given by [Danilov, 2003] in terms of both producing a stable matching and a larger number of matches on a synthetic dataset.