  In this article, we generalize a previously defined set of axioms for a
closure operation that induces balanced big Cohen-Macaulay modules. While the
original axioms were only defined in terms of finitely generated modules, these
new ones will apply to all modules over a local domain. The new axioms will
lead to a notion of phantom extensions for general modules, and we will prove
that all modules that are phantom extensions can be modified into balanced big
Cohen-Macaulay modules and are also solid modules. As a corollary, if $R$ has
characteristic $p>0$ and is $F$-finite, then all solid algebras are phantom
extensions. If $R$ also has a big test element (e.g., if $R$ is complete), then
solid algebras can be modified into balanced big Cohen-Macaulay modules.
(Hochster and Huneke have previously demonstrated that there exist solid
algebras that cannot be modified into balanced big Cohen-Macaulay algebras.) We
also point out that tight closure over local domains in characteristic $p$
generally satisfies the new axioms and that the existence of a big
Cohen-Macaulay module induces a closure operation satisfying the new axioms.
