My fear is that if I'm mass applying to other jobs, especially to startups with small teams that may not have a formal HR practice in place, a tiny set of them may call my current place for whatever reason without my knowledge. I would rather not tip off my current employer that I'm thinking of leaving; my place is a bit trigger-happy at times.
I do have colleagues in the company who will vouch for me, so references are not a problem, but there's no tactful way to put that on a resume in place of the company's name.