Storing the stream in less space
Unfortunately you will always need space $\Omega(nk)$ to support your operations.

Consider all possible contents of your stream, i.e. all subsets of $[n]\times [k]$. There are $2^{nk}$ such subsets. Take two different subsets out of this, lets say $M$ and $M'$. Since the sets are different, there is an element $(a,b)$ which is contained in $M$ but not in $M'$ or the other way around. Now we perform the operation $\mathrm{Search}(a,b)$, for which the answer should be "Yes" for $M$ and "No" for $M'$ (or vice versa). It follows that both sets need a different storage content.

Moreover, this shows that each data structure needs a different storage content for each of the subsets. It follows that each data structure needs $\Omega(nk)$ bits of space to perform your operations (even without the delete operation and without constant time querying).