  We prove that if $M$ is a rational homology sphere that is a Dehn surgery on
the Whitehead link, then $M$ is not an $L$-space if and only if $M$ supports a
coorientable taut foliation. The left orderability of some of these manifolds
is also proved, by determining which of the constructed taut foliations have
vanishing Euler class. We also present some more general results about the
structure of the $L$-space surgery slopes for links whose components are
unknotted and with pairwise linking number zero, and about the existence of
taut foliations on the fillings of a $k$-holed torus bundle over the circle
with some prescribed monodromy. Our results, combined with some results from
Roberts--Shareshian--Stein, also imply that all the rational homology spheres
that arise as integer surgeries on the Whitehead link satisfy the L-space
conjecture.
