There _is_ a "neither fish nor fowl" moment where private companies start giving out RSUs. It has to do with the overall value of the company and the size of the offer package, and rules around how much you can vest in ISOs in a given calendar year. It also has to do with the exposure to employees (and ex-employees) as shareholders pushing you above the cap that requires you to report as a public company.

At any rate, it is fairly common to get RSUs in a late-stage private company. Uber was giving out RSUs 4 years ago, I believe, and has reportedly filed for a confidential IPO as of last month.