  The purpose of this paper is to build on results in {\it{higher codimension
Iwasawa theory}}. The setting of our results involves Galois representations
arising as cyclotomic twist deformations associated to (i) the tensor product
of two cuspidal Hida families $F$ and $G$, and (ii) the tensor product of three
cuspidal Hida families $F$, $G$ and $H$. On the analytic side, we consider (i)
a pair of $3$-variable Rankin-Selberg $p$-adic $L$-functions constructed by
Hida and (ii) a balanced $4$-variable $p$-adic $L$-function (due to Hsieh and
Yamana) and an unbalanced $4$-variable $p$-adic $L$-function (whose existence
is currently conjectural). In each of these setups, when the two $p$-adic
$L$-functions generate a height two ideal in the corresponding deformation
ring, we use codimension two cycles of that ring to relate them to a pair of
pseudo-null modules.
