Minimal clique edge cover vs minimalist (assignment-minimum) ones
Consider a graph on vertex set $V_1\cup V_2\cup V_3\cup V_4\cup \{a,b,c,d\}$ where $|V_1|=|V_2|=|V_3|=|V_4|=n$. The edge set $E$ is covered by $C=\{V_1\cup\{a,c\},V_2\cup\{a,d\},V_3\cup\{b,c\},V_4\cup\{b,d\},\{a,b\},\{c,d\}\}$.
When $n$ is large enough, any minimalist cover must contain the four maximum cliques $V_1\cup\{a,c\}$ and so on, so it is not hard to show that $C$ is minimalist. However replacing the two edges $\{a,b\}$ and $\{c,d\}$ with a single clique $\{a,b,c,d\}$ results in less number of cliques.