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Casella& Berger Theorem 6.2.28: If a minimal sufficient statistics exists, any complete statistics is minimal sufficient.
So let's suppose $X_1...X_n$ are iid $Bernoulli(p)$ $p\in (0,1)$, then $\bar{X}$ is minimal sufficient statistics. But it seems that $X_1+X_2$ is complete since its distribution is Binomial(2,p), but it is not sufficient, let alone minimal sufficient...
You are completely right, and there is a lot of evident counterexamples. The word sufficient is missed in the theorem. The right statement is:
If a minimal sufficient statistics exists, any complete sufficient statistics is minimal sufficient.
