The essence of "geometric algebra" (better known as Kahler-Atiyah
algebra) is the classical Chevalley-Riesz isomorphism, which presents
the Clifford algebra of a quadratic space $(V,h)$ as a deformation
quantization of the exterior algebra of $V$. The systematic use of this
presentation allows for the automatic translation of various
computations in spin geometry into computations with differential
forms, since it identifies the Clifford bundle of a pseudo-Riemannian
manifold with its Kahler-Atiyah bundle, which is a certain deformation
quantization of the exterior bundle. This affords a particularly
'rigid' approach to certain problems in spin geometry, which is quite
useful for example in supergravity and string theory, where one often deals with various kinds of generalized Killing spinors. So this relates to a specific isomorphic presentation of Clifford algebras and Clifford bundles which is useful for certain problems. As such, it is not merely the basic theory of Clifford algebras and Clifford bundles (since it concerns certain specific realizations of such through polyvectors and forms), nor is it something radically different.

This point of view is pursued in quite some detail in the preprint:

Cortes, Lazaroiu and Shahbazi: "Spinors of real type as polyforms and
the generalized Killing equation", arXiv:1911.08658 [math.DG], https://arxiv.org/abs/1911.08658,

where the Chevalley-Riesz isomorphism is discussed in Section 3.2. Other references on the subject are:

Calin-Iuliu Lazaroiu, Elena-Mirela Babalic, Ioana-Alexandra Coman,
"Geometric algebra techniques in flux compactifications", Adv. High
Energy Phys. 2016, 7292534,
https://www.hindawi.com/journals/ahep/2016/7292534/, https://arxiv.org/abs/1212.6766

Calin-Iuliu Lazaroiu, Elena-Mirela Babalic, "Geometric algebra
techniques in flux compactifications (II)", JHEP06(2013)054,
https://link.springer.com/article/10.1007%2FJHEP06%282013%29054, https://arxiv.org/abs/1212.6918

C. I. Lazaroiu, E. M. Babalic, I. A. Coman, "The geometric algebra of
Fierz identities in arbitrary dimensions and signatures",
JHEP09(2013)156,
https://link.springer.com/article/10.1007/JHEP09(2013)156, https://arxiv.org/abs/1304.4403

A nontrivial application of this approach (the application is rigorous in that they prove some hard theorems) can be found in the papers:

Elena Mirela Babalic, Calin Iuliu Lazaroiu, "Foliated eight-manifolds
for M-theory compactification", JHEP01(2015)140,
https://link.springer.com/article/10.1007%2FJHEP01%282015%29140, https://arxiv.org/abs/1411.3148

Elena Mirela Babalic, Calin Iuliu Lazaroiu, "Singular foliations for
M-theory compactification", JHEP 03 (2015) 116,
https://link.springer.com/article/10.1007/JHEP03(2015)116, https://arxiv.org/abs/1411.3497

whose main results are summarized in these proceedings:

https://arxiv.org/abs/1503.00373

https://arxiv.org/abs/1503.00273

Lazaroiu and collaborators are string theorists, mathematical
physicists and mathematicians, so they wouldn't be interested in merely talking about Clifford algebras by another name (they do make plentiful use of Clifford algebras and bundles in their papers). Here are some examples of the kind of work which they do in spin geometry:

C. Lazaroiu, C. S. Shahbazi, "Complex Lipschitz structures and bundles of complex Clifford modules", Differential Geometry and its Applications, Vol. 61, Dec. 2018, pp. 147-169, https://www.sciencedirect.com/science/article/abs/pii/S0926224518302018?via%3Dihub

C. Lazaroiu, C. S. Shahbazi, "Real spinor bundles and real Lipschitz structures", Asian Journal of Mathematics, Vol. 23, No. 5 (2019), pp. 749-836, https://www.intlpress.com/site/pub/pages/journals/items/ajm/content/vols/0023/0005/a003/

Vicente Cortés, C. I. Lazaroiu, C. S. Shahbazi, "N=1 Geometric Supergravity and chiral triples on Riemann surfaces", Communications in Mathematical Physics volume 375, pp. 429–478(2020), https://link.springer.com/article/10.1007%2Fs00220-019-03476-7

1more comments