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.