Getting into new PhD after failing comps

Two bits of advice.

First, your SoP should be as positive as possible without hiding the setback. Talk, as usual, about your goals and how your preparation makes you a likely success.

But, I think more important, use your adviser's personal/professional contacts to help get you in the door somewhere. The advisor, even beyond a LoR, can boost your chances if they are willing to speak to others they know who can help you.

If you say "I failed comps, but ..." it is an entirely different thing than a professor saying "He failed comps, but ...". The professor is putting their reputation on the line for you. It is likely to be respected.

I agree completely with Buffy.

RE the SoP, I would add:

• You need to spend a paragraph to address it honestly. They're not going to admit you until they feel like they understand what happened, so you want to give your side of the story. In particular, you should honestly answer why you think you will be more successful someplace else.
• But don't dwell on it or make excuses. Your SoP is not the place to do a lengthy introspection or post-mortem.
• If you can summarize the problem at home in a sentence or less (e.g., "my father died"), that is probably worth doing.
• If you can take responsibility for some aspect of the failure in a sentence or less (e.g., "I didn't spend enough time studying for quantum"), that is also probably worth doing.