Algorithm to Determine if (Union of Cartesian Products of Subsets) equals (Cartesian Product of Full Sets)
The problem is coNP-complete and so unlikely to have a poly-time algorithm.  (I'm sure the observation was made before; I don't know where, but look in Garey-Johnson.)

Here is a simple reduction from 3-UNSAT $= \{\psi: \psi$ is an unsatisfiable 3-CNF$\}$.

Say $\psi(x_1, \ldots, x_t) = \bigwedge_{j = 1}^m (y_{j, 1}\vee y_{j, 2} \vee y_{j, 3})$, where
each $y_{j, a}$ is either a variable or a negated variable.

In your notation, let $A = B = \ldots = K = \{0, 1\}$, with $t$ such sets (one for each variable).  Think of each element of $S = A^t$ as an assignment to $x_1, \ldots, x_t$.  For each $j \in [m]$, create a Cartesian product set $S_j$ which describes the set of assignments that will falsify the $j^{th}$ clause.  Note that this can easily be done: we restrict precisely the 3 components corresponding to the variables appearing in this clause.

The union of the $S_j$'s equals all of $S = \{0, 1\}^t$ iff every assignment to $\psi$ falsifies some clause, i.e., iff $\psi$ is unsatisfiable.