I will focus on two disadvantages: The "But It Isn't X" Complaint (from colleagues from your home discipline) and Prejudice / Turf Defense (from colleagues from disciplines other than your home discipline).
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The "But It Isn't X" Complaint
If you apply for a job in a department in discipline X, the hiring department will care almost exclusively about your work in X. When you stand for promotion or most other sorts of disciplinary recognition, you will be evaluated almost exclusively for your work within the discipline. If you ask for your work outside the discipline to be counted equally, you will be told that it isn't really X and therefore doesn't count for much toward hiring, promotion, or recognition in discipline X.
The "But It Isn't X" Complaint is entirely understandable. Shouldn't hiring and promotion into a philosophy department, for example, and recognition in philosophy, depend on the candidate's contributions to philosophy? And even if in principle the evaluators want work outside of their discipline to count equally, they will feel unable to properly evaluate it. In the disciplinary evaluations on which most of academia is built, within-discipline contributions count most.
The almost inevitable consequence is that researchers who devote substantial time to interdisciplinary work will be severely disadvantaged in hiring, promotion, and disciplinary recognition.
What to Do on Behalf of Your Interdisciplinary Colleagues
If a department wants to recognize interdisciplinary colleagues appropriately in hiring, promotion, and other types of evaluation, they need to ask not "how much has this person contributed to our discipline?" but rather (1.) "how much has this person contributed to academia as a whole?" and (2.) "has this person contributed enough to our discipline to still count as member of this discipline?" Suppose someone straddles two disciplines 50/50, and over some period of time they publish three excellent papers in their home discipline and three excellent papers in another discipline. Evaluate them not according to the three home-discipline papers, with the three others as "frosting", but treat all six papers on a par. Of course, if the majority of papers are in another discipline, at some point it would be reasonable to consider a change of department. But until that time, all contributions should be valued and evaluated by the home department.
If there is to be room in academia, as I think there should be, for people who bridge two disciplines, those people need to be valued for their contributions to both disciplines. If people in their home discipline cannot expertly evaluate that person's interdisciplinary work -- quite understandable! -- they should consult with others from the appropriate discipline, or even better with others who are interdisciplinary between the same two disciplines.
What to Do If You Are the Interdisciplinary Researcher
Assuming your colleagues and evaluators are not implementing the strategy above, as most will not, I advise three strategies:
(1.) Do as much in your home discipline as your colleagues do. Publish the six papers in your home discipline and three outside your discipline. This isn't easy to implement, of course! But one of the advantages of interdisciplinary research is that your expertise outside of your home discipline can be a font of fresh ideas. If your c.v. contains as much good work in X as your colleagues', it doesn't matter so much if they think of your other work as of secondary importance.
(2.) Take advantage of higher-level administrators' appreciation of interdisciplinarity. In my experience, the majority of higher-level administrators (deans, etc.) value interdisciplinarity. Their evaluations rarely matter enough to compensate for the structural disadvantages I mentioned above, but often their evaluations matter somewhat. There are sometimes grant opportunities, teaching release opportunities, or other recognition for interdisciplinary work; keep your eyes open for these. Also, if your disciplinary colleagues are supportive, you can remind them that there are aspects of your research profile that will be attractive to administrators because of your interdisciplinarity. This can lead your colleagues' to be more assertive in making your case than they would otherwise be, anticipating approval from the higher-ups.
(3.) Relabel your work as a contribution to your discipline. This is the boldest move, and it will have mixed success at best. For example, when I first started doing work in psychology I thought of it just as work in psychology that had consequences for philosophy. After all, if you run an empirical experiment that looks like a psychology experiment, or contribute an article to a psychology journal, isn't that doing psychology?
However, starting around 2003, Joshua Knobe, Shaun Nichols, Steve Stich, Jonathan Weinberg, and others started calling their empirical research on non-philosophers' philosophical opinions "experimental philosophy" rather than psychology, and they successfully campaigned to publish some of it in straight-up philosophy journals. They partly succeeded in changing the borders of the discipline of philosophy to include work that would previously have been called psychology. They haven't convinced everyone, of course. Not all philosophers think of "experimental philosophy" as really philosophy. But the situation is better than it was. Arguably, "behavioral economics" is a similar story, even more successful.
Similarly, I have been starting to make the case that writing fiction can also be a way of doing philosophy -- witness Rousseau, Sartre, Nietzsche, Plato, Murdoch, Voltaire, etc.!
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Prejudice / Turf Defense
Interdisciplinary prejudice and turf defense are slightly different but related phenomena.
Interdisciplinary prejudice is the understandable default assumption that someone outside of your discipline isn't going to be nearly as good at work in your discipline as someone whose formal affiliation and training is in your discipline. Turf defense is an emotional reaction to the threatening idea that someone outside of your discipline might be as good as you and your colleagues, or better, at work in your own discipline [ETA:] or that others might falsely perceive them that way.
Usually, interdisciplinary prejudice is justified, and perhaps not deserving of a pejorative label. If a non-philosopher submits something to a philosophy journal, odds are good that it won't be an excellent work of philosophy. If a philosopher tries to run a psychology experiment, odds are good that their methods and analyses won't be as solid as a psychologist's. For similar reasons, turf defense isn't wholly unjustified: You don't want others to mistakenly think that the non-X researcher's probably-inferior work is as good as a disciplinary expert's work, so it makes sense in a way to guard against incursions. The turf defense reaction is also, I think, partly driven by feelings that the outsider is being disrespectful: If an outsider thinks they can come in and beat us at our own game, that seems to suggest that they lack respect for our years of hard work and disciplinary training.
However, sometimes people really can do excellent work in more than one discipline. It takes years of effort to acquire the knowledge and skills; but people do sometimes put in the requisite time and effort. An Associate Professor of X with a strong interdisciplinary focus might have as much knowledge of and experience in Y as an Assistant Professor in Y. (It would be almost superhuman, though, for an Associate Professor in X to have as much knowledge and experience in Y as an Associate Professor in Y, unless the situation is very unusual.) However, even when the outsider does have the requisite knowledge and skills, it is, I fear, a sociological fact that substantial prejudice and turf defense remain.
What to Do on Behalf of Interdisciplinary Colleagues
(1.) Be aware of your possible interdisciplinary prejudice and turf defense and the fact that they are not always justified. Try to evaluate work in your home discipline by someone outside of your home discipline in approximately the same way you would evaluate other contributions to your discipline.
(2.) Implement anonymous review when possible. If the work passes muster, it shouldn't matter if it's from a Stanford professor in your discipline or someone from a less prestigious university with a different disciplinary affiliation or from a construction worker in Tallahassee.
(3.) Where anonymous review isn't possible, downplay departmental affiliations on the first page of articles and applications -- for the same reasons.
What to Do If You Are the Interdisciplinary Researcher
(1.) Collaborate with someone from the other discipline. There appears to be much less prejudice and turf defense when at least one member of the research team is from the target discipline. Furthermore, the collaborator will bring an inside-the-discipline perspective that it is very difficult to achieve from outside a discipline, even if one has substantial expertise.
(2.) Watch for shibboleths. By shibboleths I mean superficial signs of being an insider rather than an outsider. It helps reduce prejudice and turf defense the more you can write and speak indistinguishably from members of the target discipline. (Collaborators can help with this.) If you sound like an outsider, even if your content is good, that will tend to amplify negative reactions to your work.
(3.) Cite thoroughly and carefully early in your project. Show, from the very beginning, thorough and serious engagement with the existing work in the target discipline. This shows respect for that work, reducing the turf defense reaction, and it shows that you have substantial expertise, reducing the prejudice reaction.
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Barring radical changes, structural disadvantages will continue to impair people who do interdisciplinary work. However, I do also believe that there is one major compensatory advantage, over the long run of a research career. Often, the freshest and most fruitful academic ideas come from researchers with expertise in more than one area, who can use their expertise in Y to shine new interesting light on X. Your colleagues won't always appreciate this right away. But in the long run, you will have different things to say than those whose expertise is exclusively within a single discipline. You will have a distinctive perspective and contribution.