Let $X$ be the product $Gr_i(V)\times Gr_j(V)$ of two Grassmannians where $V$ is a complex vector space of dimension $d$. There is an open $U\subset X$ formed by all those $(V',V'')\in X$ such that $V'$ and $V''$ intersect transversally, i.e., $\dim V'\cap V''=d-i-j$.

Assume $d-i-j>0$. There is a natural map $f:U\to Gr_{d-i-j}(V)$ given as $(V',V'')\mapsto V'\cap V''$. There exists a blow-up $\tilde X\to X$ whith center $\subset X\setminus U$ and such that $f$ extends to $\tilde X$ (see e.g. Hartshorne, chapter 2, example 7.17.3). There is a vector bundle on $U$ formed by "intersecting" the pullbacks of tautological bundles on $Gr_i(V)$ and $Gr_j(V)$ to $X$, and this bundle extends to $\tilde X$.

Such an $\tilde X$ looks like a natural object to consider and it was probably considered before. However, I wasn't able to find any references, and I would like to ask if anyone knows any. In particular I'm interested in the following:

How can one represent $\tilde X\to X$ as a sequence of blow-ups with smooth centres, or at worst, with centres which are regularly embedded subschemes?

Have the (say rational or integral) cohomology of $\tilde X$ and/or the Chow ring been computed?

Of course, $\tilde X$ is not unique: there are many ways to blow up $X$ so that $f$ extends. For my purposes any such $\tilde X$ would do.