Pseudo orthogonal Latin squares

Two Latin squares $A,B$ of order $n$ are called pseudo orthogonal if for any $1\le i,j\le n$ there exists a $k,1\le k\le n$, such that $A(i,k)=B(j,k)$. We prove that the existence of a family of $m$ mutually pseudo orthogonal Latin squares of order $n$ is equivalent to the existence of a family of $m$ mutually orthogonal Latin squares of order $n$. We also obtain exact values of clique partition numbers of several classes of complete multipartite graphs and of the tensor product of complete graphs.