I know that there are many examples of spaces $Y\subset X$ such that $X$ and $Y$ are homotopy equivalent but there is no deformation retract of $X$ to $Y$ (e.g., Does homotopy equivalence to a subspace imply (weak) deformation retract?). However, all these examples have some bad properties that analytic spaces do not have (or simply are not a good example).
Given a real analytic space $X$ and a real analytic subspace $Y$, is it true that there is a deformation retract of $X$ into $Y$ if the inclusion of $Y$ in $X$ is a homotopy equivalence?
In my particular problem I take $X$ a complex variety in $\mathbb{C}^n$ and $Y=X\cap\mathbb{R}^n$; and I have something more: the inclusion is also an isomorphism in all homotopy groups. Maybe some real-algebraic geometry theorem solves this (?)