I have worked in academia, at the research center of a telecommunications company (Tellabs), and at two different FFRDCs (MIT Lincoln Laboratory and IDA). At all of the non-academic jobs, I have done "real math," published papers, attended conferences, given talks, etc. So it is certainly possible to continue doing "real math" outside of academia.
You should be aware, however, that in almost any non-academic job, there is pressure on you to produce results that are "useful" for the company or the government. The amount of such pressure varies, but it always exists, because ultimately that is the main justification for your paycheck. In academia, the corresponding fact is that in almost any academic job, there is pressure on you to teach, since that is usually the justification for a significant portion of your salary. Finding a non-academic job where there is no pressure on you to do anything "useful" is akin to finding an academic job where you have no teaching responsibilities.
Certain high-tech companies and certain FFRDC's recognize that a good way to attract top talent is to give their employees the freedom to pursue their own research interests, whatever that may be. All the non-academic jobs I had were like this. They actively encouraged me to spend some amount of my time doing "real math" regardless of whether the results were of any "use." How much time? Well, if the company was doing well, and if I was doing a good job of producing "useful" results that they liked, then they would give me more freedom. But if the company was doing poorly then they would start to squeeze. During the telecom industry meltdown in the late 1990s, Tellabs eventually eliminated its research center entirely, along with my job; Bell Labs (more famously) suffered a similar fate.
So far I have been drawing a dichotomy between "what the company finds useful" and "real math," and maybe you don't find that satisfactory. After all, if you're sufficiently motivated, you can do "real math" on your own time regardless of what your "day job" is. Maybe what you want is a job where providing what is useful to the company involves doing real math. This is a taller order; for example, at Lincoln Labs I found that there was almost no real math involved in the work they wanted me to do, and I eventually left that job for that reason even though it was a great job in almost every other respect. However, it is still possible to find such jobs, depending on what area of math you are interested in. If you are interested in large cardinals and are hoping for a job where your theorems about large cardinals will be "useful" then you are probably out of luck. However, if your interests lean towards areas with known relevance to computer science or various branches of engineering then your chances are much better. The NSA scores pretty well in this regard since it is no secret that number theory and various other branches of so-called "pure" mathematics are relevant to cryptology.
In summary, jobs where you do "real math" do exist. When considering such a job, though, you should first ask yourself, will I enjoy producing what this company considers to be "useful" results? If the answer is no, then you will probably not be happy at the job even if they give you some freedom to do "real math." However, if the answer is yes, and the company gives you some amount of freedom to do "real math," then it will probably be an excellent fit for you.
I can comment from personal experience that my wife lost her job teaching Physics at a private 4-year (for the undergraduate degree), not-for-profit, accreditted long-standing (ancient) university in the USA (yay, Boston) with good reputation, despite her excellent teaching reviews from her undergraduate students. They did not renew her contract in the fall just before the Accreditation Review committee was due to visit the University.
Details: she had a terminal Master of Science in Physics, followed by ten years in the defense industry, followed by a Ph.D. in a different field. She taught as an adjunct for a few years before being offered a real position, but was let go because the perception that a Masters level in the field being taught would be seen as a negative by the accrediting board, even with a Ph.D. in a "neighboring" hard science field. She tried to argue for her position to no avail, as they said that they could not risk losing their accreditation. She did not want to take a step back down to teaching as an Adjunct Professor at the same institution again, so they parted ways.
It may be very difficult for your friend to get hired with a Ph.D. in an "outside" field. It may not be just; but that is what happened. They continue to use adjuncts to teach $>70$% of their class-load for undergraduates.
Of course, I must add that you should always apply for a job for which you think you are qualified.
Best Answer
Having been on both sides of the issue, I might say that having considered it for some time, I really don't know! But in reality if you are looking for a position at a research university, the Dean will want to have evidence (or the non-research faculty will want to have evidence) that you care about teaching. More precisely, some subset of your peers might have a very specific teaching philosophy although they may not be able to articulate it. Those peers want to know if your teaching philosophy coincides with theirs.
A few years back everyone was "hot" on the use of technology in the classroom. I don't know what that means, but suppose that it means using TI calculators, power point (the horror, the horror) or a course blog. If you have a point of view on the positive value of these things then you should say so.
The problem is that each department has its own mix of bozos. I am pretty much a chalk on slate kind of guy, and when someone tells me they like clickers in large classes, I wonder do they turn around to look at their students faces. So in an ideal world you would tailor your teaching statement to the place you want to go, or to the place that you are applying. Of course, you don't want to write 200 teaching statements, so that won't work.
So I am back to the original premise. They want to know that you have thought about teaching.