Find a Minimal Set of Combined Tuples
I am not aware of whether this problem has a name or not, but it can be shown that it is NP-complete by reduction from the maximum independent subset problem:

Given an undirected graph of $N$ vertices and $M$ edges, we can map the nodes to a set of $M$-tuples $\{x_1,...,x_n\}$ such that any two tuples are "compatible" if and only if the corresponding nodes do not share an edge:


Given $i$-th undirected edge $\{a,b\}$ with $ a\leq b$ we set $x_{a}[i]$ to 1 and $x_{b}[i]$ to 2.
Any element that remains unset by the previous rule is set to #.


Now:


For each pair of nodes that share an edge, the corresponding tuples will not be compatible because they have different values in the position corresponding to the edge.
For each pair of nodes that do not share an edge, for each position in the tuples at least one of them has # assigned in that position. Hence,they will be compatible.


By definition:


An independent subset of the graph is such that it does not contain two nodes that share an edge.
A compatible subset of tuples is such that it does not contain two tuples that are not compatible.


Hence a subset of nodes is independent if and only if the corresponding subset of tuples is compatible. Then, the size of the maximum independent set in the original graph is equal to the size of the maximum set of compatible tuples.