How to handle not having my PhD advisor as a reference?

I've personally seen a few careers survive this exact problem (caveat: I don't necessarily see the ones that don't), but it isn't easy.

The best way to surmount this problem is to be referred into a job in an industry/government/NGO lab - and have someone within the organization pulling for you. If you're doing an academic post-doc right now, that's fine, but you need to start connecting to people who work in the kinds of places you want to work. If you do another post-doc, do one for a company or a government lab. If you've got someone who can pull you into the fold at a company or lab, you've got no issue at all.

Here's my advice if you are sending out cold-call applications:

The issue won't come up in the application stage (don't mention anything negative about your experience in your cover letter, or your application will get trashed). The trick will be how to handle yourself during the interview stage.

You say here:

"in retrospect I see that I could have avoided a breakdown in our relationship if I had been more deferential."

You're going to need a much better story than that. Because you are going to have to explain it at some point (probably when someone asks why you haven't listed your old PI as a reference), and that line will throw up red flags about your ability to handle being managed. You'll have to be prepared to talk about it in some detail, but focus everything on what you learned from it. How do you handle conflict better now? Have recent examples ready of your excellent interpersonal skills.

Avoid talking about this problem for more than a few minutes in an interview, but don't hide the fact that you've have this issue in the past. Employers will call your references, and chances are good that it will come up. Even if one of your references mentions it with you in the best possible light, if it's the first time the employer has heard of it they will feel like you've withheld information. While companies will generally not call someone you haven't listed as a reference, they will get in touch with everyone they know who may have also worked for your old boss, gotten a degree from your old department, collaborated with your old group. This may be an issue if your field of study is small. If you've changed your research focus somewhat since then, this is less likely to happen.

You can overcome this, but BE PREPARED. And once you get your first non-postdoc job, it'll all be downhill from there.

Here in Germany, this wouldn't be a problem, as the PhD adviser is normally excluded from providing references for hiring processes in academia, since it's assumed the advisor would support the student.

However, while there is reason to be concerned about your advisor not giving you a good letter of recommendation, all is not lost. Since this is not your previous position (you're coming from a postdoc), the PhD advisor's weight will not be as significant. It's still important, and you better have a very good "elevator speech" explanation why he might not support you. But the fact that you have a postdoctoral advisor means that you do have some credentials; if he can write a strong letter in support, it might further give credence to the idea that your there was just dysfunction between you and your PhD advisor.

There are some good answers here. I would like to add that this happened to me also. For a long time, I was also apprehensive and worried about prospective employers noticing that I don't list my advisor as reference, how to defend myself in interviews etc. but then I realized the best solution is to be upfront about it. When you are hiding something, people think you are guilty vs. when you are upfront about it - people tend to sympathize, plus most importantly you have no fear.