Subanalytic Bundles and Tubular Neighbourhoods of Zero-Loci

We introduce the natural and fairly general notion of a subanalytic bundle (with a finite dimensional vector space $P$ of sections) on a subanalytic subset $X$ of a real analytic manifold $M$, and prove that when $M$ is compact, there is a Baire subset $U$ of sections in $P$ whose zero-loci in $X$ have tubular neighbourhoods, homeomorphic to the restriction of the given bundle to these zero-loci.