  Given a monomial $m$ in a polynomial ring and a subset $L$ of the variables
of the polynomial ring, the principal $L$-Borel ideal generated by $m$ is the
ideal generated by all monomials which can be obtained from $m$ by successively
replacing 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 $L_i$-Borel
for $i=1,\ldots,r$ (where the subsets $L_1,\ldots,L_r$ may be different for
each ideal), we prove in essence that if the bipartite incidence graph among
the subsets $L_1,\ldots,L_r$ is chordal bipartite, then the defining equations
of the multi-Rees algebra of $\mathcal{I}$ has a Gr\"obner basis of quadrics
with squarefree lead terms under lexicographic order. Thus the multi-Rees
algebra of such a collection of ideals is Koszul, Cohen-Macaulay, and normal.
This significantly generalizes a theorem of Ohsugi and Hibi on Koszul bipartite
graphs. As a corollary we obtain that the multi-Rees algebra of a collection of
principal Borel ideals is Koszul. To prove our main result we use a fiber-wise
Gr\"obner basis criterion for the kernel of a toric map and we introduce a
modification of Sturmfels' sorting algorithm.
