I think this question is a bit too broad, as it depends on the theory you're talking about.

Lubos Motl has just posted a new article about (a similar to this) idea on TRF.

Well, you do have spacetime in string theory, for example. But as Lubos Motl mentions in the article linked, the metric tensor isn't exactly that well-defined at the stringy scales. To quote the article:

However, quantum gravity doesn't allow you things like that. The metric tensor is only good and well-defined in an effective description of quantum gravity. At shorter distances, it just ceases to be a good observable. Well-defined observables in quantum gravity are different; the gauge fields in the $\mathcal N=4$ Yang-Mills theory involved in the most famous example of the AdS/CFT correspondence are an example. The matrices $X,P,Θ$ in Matrix Theory are another example.

Same with things like Twistor theory, where the real space is the space of twistors.

In LQG, though, there is indeed a vierbin; though.