  We determine Gr\"{o}bner bases for the presentation ideals of multi-Rees
algebras and their special fiber rings. Specifically, we focus on modules that
are direct sums of ideals generated by either maximal minors of a two-sided
ladder matrix or unit interval determinantal ideals. Our analysis reveals that
the multi-blowup algebras are Koszul Cohen--Macaulay normal domains, possess
rational singularities in characteristic zero, and are F-rational in positive
characteristic.
