This is too long for a comment.
I found the article
Lazaroiu, Calin Iuliu; Babalic, Elena Mirela; Coman, Ioana Alexandra, Geometric algebra techniques in flux compactifications, Adv. High Energy Phys. 2016, Article ID 7292534, 42 p. (2016). ZBL1366.83098.
on the classification of Killing (s)pinors using geometric algebra with applications to $\mathcal N=1$ M-theory compactifications to 3D.
Their perspective on geometric algebra is explained in section 3. The central object seems to be what they call the Kaehler-Atiyah algebra over some (pseudo)Riemannian manifold $M$ which as far as I can tell will reduce to geometric algebra as in Wikipedia when $M$ is Minkowski space. They also sketch how the KA algebra is obtained by a quantization procedure.
EDIT: In light of the sociological comments in the third paragraph in the question I should point out that I don't think the authors of that paper would say they are in the group of people who identify as "GA" (Lazaroiu, whose work I've read before, is a string theorist)