In theorem 1.2 of Brian Conrad's handout [Operations with Pseudo-Riemannian metrics](http://math.stanford.edu/%7Econrad/diffgeomPage/handouts/metricops.pdf), the author writes



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> **Theorem 1.2.** Every $C^p$ vector bundle $E\to M$ over a $C^p$ manifold with corners $0\leq p\leq \infty$ admits a Riemannian metric. In particular, every smooth manifold with corners admits a structure of a Riemannian manifold with corners.
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> The main tool in the proof will be $C^p$ partitions of unity, so it is amazing that Morrey and Grauert proved that every real-analytic vector bundle over a real-analytic manifold admits a real-analytic Riemannian metric tensor. The real-analytic case rests on serious input from the theory of several complex variables in order to circumvent the lack of real-analytic partitions of unity.
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What is the idea of Morrey and Grauert's proof? What's the formalism that allows circumventing partitions of unity? What are the miracles of complex analysis that make things work?



Morrey & Grauert - real analytic vector bundles admits analytic Riemannian metric
I'm not sure this should be an answer, but here is a proof that might be more conceptual resting on one non-trivial fact: if $E\rightarrow M$ is a real analytic fiber bundle over a real analytic manifold $M$ which admits a continuous section, then $E\rightarrow M$ admits a real analytic section. 


If you take this one black box, the proof is identical to the standard "reduction of structure" interpretation of a metric on a vector bundle. Specifically, let $V\rightarrow M$ be a real analytic vector bundle. If $E\rightarrow M$ is the principal $GL(n,\mathbb{R})$-frame bundle of $V$, it has an induced real analytic structure. A $C^{r}$-metric on $V$ is equivalent to a $C^{r}$-reduction of structure of $E$ to the orthogonal group, which is equivalent to a $C^{r}$-section of the associated bundle of homogeneous spaces $E\times\_{GL(n,\mathbb{R})}GL(n, \mathbb{R})/O(n).$ 


This bundle of homogeneous spaces has a $C^{0}$-section since the fibers are contractible, and therefore by our black box it has a real analytic section. This proves $V\rightarrow M$ admits a real analytic metric.


See <https://www.math.uni-augsburg.de/prof/geo/Dokumente/stein106.pdf> section 5.8 for a discussion of these matters. There they verify that one can even approximate as well as one likes a $C^{k}$ metric by an analytic one in the $C^{k}$-topology. Of course, as you see there, the proofs of the black box rely
on embedding $M$ real analytically into some Euclidean space, and therefore this proof is really not "different" than the one explained by Mike Miller in the comments. 


