Receiving offer contingent on speaking with current supervisor

I was recently moved to a new position, 3 months ago, and my relationship with the current supervisor is so poor that I decided to leave. I have gone through the interview process at another employer, and on the application, I checked the box that they cannot contact my current employer(it is ridiculous that the box was even there if contacting your current supervisor is a requirement to be hired). Now I have been told by my prospective employer that I have been chosen, but they cannot extend an offer until they contact my current supervisor and current company. I hate to walk away from this new job, but I cannot see a way that I can allow the prospective employer to contact my current supervisor while I am still employed there or even after I have put in 2 weeks notice. Even if they extend an offer and I accept it, if they contact my current supervisor I am toast. My company gives out full references, not the standard, dates of employment only BS. They know it creates liability for them but they do it anyway. Has anyone ever run into a situation like this with a prospective employer? Any ideas on how I can handle this? I really want the new job but don’t see how I can let them contact my current employer.

This is the first employer that I have ever heard of that is insisting on contacting “current supervisor” before they even make an offer to me. In essence they want me to give 2 weeks notice because they said they will give me an offer, then they want to call my current supervisor, then if all is cool after that, they will give me a written offer, which by the way I haven’t even negotiated with them, so I guess this whole BS procedure gives them all the negotiating leverage too. In essence I would have to take whatever offer they have if all goes well or face unemployment. Because i am gone once my supervisor knows I am getting an offer possibly somewhere else.

Every other job change in my life, you just check “don’t contact current employer” and all is good, they call employers prior to your current employer

In my mind, I was gonna walk from this offer because their “ask” just involves too much risk. There are plenty of employers that don’t do this, pretty much 99% of all other employers don’t do this, so I won’t be stuck in my current job forever, but I was just searching for a way around this, because I really want the new job.