Can a theory with plural quantification have elementary equivalent structures that use different length sequences to satisfy sentences?

Question

Suppose we have a language $\mathcal{L}$ that is first order but allows for plural quantification, and let $T$ be an $\mathcal{L}$-theory that includes a sentence of the form $$ \exists\bar{x}\;\phi(\bar{x})$$ Now, suppose we have structure, $\mathcal{M}\models T$, and $\bar{x}\in M^a$ such that $\phi(\bar{x})$. Is it possible to have $\mathcal{N}\models T$ that is elementary equivalent to $\mathcal{M}$ such that for any $\bar{y}\in N^b$ satisfying $\phi(\bar{y})$ it follows that $a\neq b$? In other words, is it possible for there to be two structures satisfying the same conditions, but using different amounts of variables to do so?