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Given a monomial $m$ in a polynomial ring and a subset $L$ of the variablesof the polynomial ring, the principal $L$-Borel ideal generated by $m$ is theideal generated by all monomials which can be obtained from $m$ by successivelyreplacing variables of $m$ by those which are in $L$ and have smaller index.Given a collection $\mathcal{I}=\{I_1,\ldots,I_r\}$ where $I_i$ is $L_i$-Borelfor $i=1,\ldots,r$ (where the subsets $L_1,\ldots,L_r$ may be different foreach ideal), we prove in essence that if the bipartite incidence graph amongthe subsets $L_1,\ldots,L_r$ is chordal bipartite, then the defining equationsof the multi-Rees algebra of $\mathcal{I}$ has a Gr\"obner basis of quadricswith squarefree lead terms under lexicographic order. Thus the multi-Reesalgebra of such a collection of ideals is Koszul, Cohen-Macaulay, and normal.This significantly generalizes a theorem of Ohsugi and Hibi on Koszul bipartitegraphs. As a corollary we obtain that the multi-Rees algebra of a collection ofprincipal Borel ideals is Koszul. To prove our main result we use a fiber-wiseGr\"obner basis criterion for the kernel of a toric map and we introduce amodification of Sturmfels' sorting algorithm.

Koszul multi-Rees algebras of principal $L$-Borel Ideals