Postdoc before PhD completion
Groups seeking postdocs will absolutely consider you as an applicant before you have completed your PhD. They will, however, want to be certain that you have completed the entire process and all its requirements prior to officially beginning your new job, as the comments above have said. Be as honest with them as possible about when you expect to finish.
Universities will vary in terms of how they expect you to prove you've completed your PhD: some will accept official letters while others will want a photocopy of your physical degree (I was asked for the latter and found it rather, er, surprising). Luckily, none of that is likely to happen during an interview.
Apply now!!! In my experience (sciences in US), postdocs are usually interviewed fairly far before they finish. I think I applied in December, interviewed in February, defended that summer, started the next fall. (This was a pretty early start, and it's possible to go faster, so don't panic yet.) Getting a postdoc offer also serves as pretty strong encouragement to your advisor to ensure you can defend and leave in a timely manner.
If you can defend prior to your funding running out, that would obviously be the best. However, there are other options:
- Your advisor finds more funding - nearly as good.
- You find a nonacademic job, and finish your thesis in your spare time. Possible but not great (common in humanities PhDs, which is why they can take ten years).
- You could start with your postdoc group prior to graduating - they might be able to hire you as a "technician" before you finish. You could then write up your thesis in your spare time. Some groups will be OK with this, since they get a PD-level person to start out at a lower salary - but others will object to not having 100% of your focus, or will have grant funding that isn't flexible on this.
Options 2 and 3 are going to be much much easier if you don't have much scientific work to finish - i.e. you mostly need to write up and defend, with maybe some additional computation/analysis. They will be near-impossible if you have additional in-lab work to finish. (I do know of one unusual case with a super nice postdoctoral advisor, where the student finished some of their thesis work in their new advisor's lab. This is very much the exception to the rule.)
would universities not consider you until you have completed the PhD
You should take a look at advertisements about postdoc/researcher positions to have an idea. What I have seen so far is only "close to finish a PhD".
I started applying for postdoc as soon as my advisor said my contribution was enough to graduate. However, I kept track of open positions long long before that.
I did onsite interviews with 2 groups, and got one offer before started writing my thesis. At that time, I was so scared that I would blow my chance, so I nervously asked this question, but everything was OK.