Is pursuing a second PhD in the same or related field a viable means of returning to academia after a decade in industry?

In order to return to academia, you will need to satisfy certain requirements.

• Most academic jobs require holding a Ph.D. in the relevant field. You already have one. A second Ph.D. won't help you here.
• Teaching jobs require demonstrated teaching experience. A second Ph.D. won't help you here. You could certainly try to build a track record of successful teaching by getting in touch with a local community college and offering to teach a few courses - possibly even while holding your industry day job. If you are serious about returning to academia, this would be an investment, not a source of income, so you could offer to teach for little or no remuneration.
• Research (and teaching) jobs require publications. A second Ph.D. won't help you here. Instead, get up to speed on current research, find a good open problem, work on it and publish some papers.
• Finally, contacts will be invaluable. A second Ph.D. won't help you here. Instead, attend conferences, give presentations, get to know people, leave a good impression and hint that you would like to return to academia.

Note the common thread going through all these points: a second Ph.D. in the same field won't help you. Instead, invest your time in building the kind of portfolio (teaching, publications, contacts) that you do need.

This will likely be an uphill battle, but if you are motivated enough, then go for it. Good luck!

I think it will be very difficult -- perhaps prohibitively difficult -- to get a PhD in mathematics (or mathematical physics, which is all but mathematics) in the US given that you already have a PhD in mathematics from the US. It came up within recent memory in the math PhD program at my university that we had an applicant (from outside the US) who already had a PhD. We went so far as inquiring to the graduate school whether it would be against the rules to admit such a student. It wasn't. Nevertheless, suffice it to say that the student was not admitted to the program, and if this happened again I highly doubt the outcome would be different.

Maybe it will help to look at it from our perspective. We are trying to train future academics. (It is true that some PhDs leave academia, but for graduates of my department this is a minority: probably less than 1/3. Moreover, no matter what they go on to do, it is only fair to say that we are training students for academic jobs.) We want to spend our limited resources on people that want to say in academia and are capable of doing so. Someone who wants to get another math PhD ten years later is getting the horns of that dilemma: if you are actually capable of continuing on in academic mathematical career, then one math PhD ought to be sufficient. A US math PhD is not meant to be merely an apprenticeship in a certain specialized subfield of mathematics: it is meant to train you to do teach yourself new mathematics and do research independently. By attempting to enroll in a second PhD program, you are signalling that you didn't acquire these skills the first time around....or maybe you did but chose not to exercise them for so long that they atrophied. Either way makes you a poor candidate.

What can you do? I would have to say that getting a PhD and spending a decade away from mathematical research and teaching is closer to a "lockout situation" than most others I can think of. If I'm honest, I don't completely understand the story as you frame it: you write

I have a PhD from a reasonably good program in pure math, but I had no support from my department or adviser and so went into industry afterward.

You lost me: awarding someone a PhD is a highly non-negligible level of support. If you have a PhD from a reasonably good program in pure math, then you can get some kind of academic job: maybe an adjunct position, but some kind of job. You chose not to and went with that choice for ten years. If you really loved academia, wouldn't you have done some academic activities in that time? I'm afraid it sounds more like: you don't like either academia or industry. This is not a contradiction: there are more things you can do than that!

But let me give you the benefit of the doubt that for whatever reason you spent ten years away from the thing that you really want to do. What can you do? Well, what I said before about having a PhD getting you some kind of academic job is still true: in fact, at certain institutions, "even" in pure math your industry experience will be a big positive. A community college teacher may be a "pure mathematician" in her own private time, but those skills are very unlikely to be drawn upon in her work. Someone with industry connections of any sort would be preferable.

So I think you claw your way back into academia by taking a job at a community or regional four-year college: quite possibly a temporary job. If you play it right, you might be able to do this for a time while still keeping your industry job: I certainly recommend this. After you build up a teaching portfolio, the ten-year gap in your CV will recede into the background, and -- assuming you do well, work hard, and so forth -- you should eventually be able to land a permanent teaching position somewhere.

If you want to break back into mathematical research: go ahead and do it, you don't need to leave your job for that. Go to some conferences, read some papers, start contacting people in your intended field...just do it. It will be slow going at first, but because you already have a PhD in mathematics you do have the capacity to learn and acquire mathematics. So learn and acquire mathematics. Have fun...