The well-known connection between hyperidentities of an algebra and identities satisfied by the clone of this algebra is studied here in a restricted setting, that of $n$-ary strongly full hyperidentities and identities of the $n$-ary clone of term operations of an algebra induced by strongly full terms, both of a type consisting only of $n$-ary operation symbols. We call such a type an $n$-ary type. Using the concept of a weakly invariant congruence relation we characterize varieties of $n$-ary type whose identities consist of strongly full terms which are closed under taking of isomorphic copies of their clones of all strongly full $n$-ary term operations. Finally, we show that a variety of $n$-ary type defined by identities consisting of strongly full terms has this property if and only if it is $\mathcal O_{SF}$-solid for the monoid $\mathcal O_{SF}$ of all strongly full hypersubstitutions which have surjective extensions.