Is it commonplace for a recruiter to ask to see an unpublished PhD thesis?

In a word, yes. It is very common for academic employers to want to know about a candidate's research in progress, and they often ask for research plans, unpublished manuscripts, reports on ongoing projects, etc. From the employer's point of view, they want to know as much as possible about what the candidate is doing, so as to evaluate the promise of their research program and their productivity. This is especially true for junior candidates who do not already have a large body of published work. So a request for a draft of a PhD thesis would not be out of line.

When a candidate shares such material as part of their application, the hiring professor or committee has an ethical obligation to hold it in confidence. They should not circulate it beyond those people within the department who are involved in the hiring decision. Also, it would be ethically inappropriate for anyone with access to this material to exploit it for the gain of their own research program (e.g. by trying to solve the candidate's thesis problem before they do, or giving it to one of their own students). As the candidate, you have the right to expect that this will not happen.

Of course, as a matter of practice, if you want the job, you don't have much choice but to give them what they ask. But I wouldn't see such a request as unusual or unreasonable, and I don't think you need worry about them using it unfairly. If you are still worried, you could send them the thesis along with a note saying "since this is work in progress, I would ask that you keep it in confidence".

Also, I would say it's an exaggeration to say a thesis should be a "closely guarded secret" or to react with "horror" to a request to share it. It's generally prudent not to share unpublished work indiscriminately, but it's not as if it were missile launch codes. If there is something to be gained by sharing it with someone (e.g. useful input from an expert, a potential collaborator, a job) then often that's a good idea. It seems to be pretty common for people starting out in academia to overestimate the risks of people stealing their work: yes, there are horror stories, but in the long run, you usually have more to lose from excessive secrecy than from reasonable openness. Paranoia is generally not a helpful trait for an academic.

It might depend on the field, but it strikes me as pretty normal to ask for the PhD thesis in the context of an academic job application.

Your PhD thesis shows the quality of your research. Your PhD thesis shows the quality of your ability to communicate your research. Both are essential skills in (academic) research. Unless you already have many published papers — and it appears that you do not (or else why would it be secret?) — then your PhD thesis is the only document that can serve as evidence that you do possess those skills.

If you are worried about results leaking, you can ask for the manuscript to be treated confidentially. If you are worried about the recruiter stealing and abusing your results, you might want to reconsider if you want to work there in the first place.

As others have said, it's not unusual. However, to answer those talking about paranoia; at my Uni, we used to have seminars where we discussed with other PhD students different aspects of our thesis. This was done with the explicit agreement that these discussion will remain in strict confidentiality, and we will not use each other's work. However, there were no written, signed documents to ensure this. So after one such sessions where I was explaining a central point of my thesis, one of the fellow students, who was researching an entirely different subject, was quite interested and stayed on to discuss at length my thesis. I was flattered by the attention - only to find, a few months later, that he published a book containing, basically, all of my PhD thesis. I had to completely change my thesis, and although my tutor commented on the fact that the book contained what looked like my work, there was nothing I could do. The guy who stole my work is now a lecturer at the same University.