I am looking for a counter example which shows, that a full rational 2D CFT (with respect to a given chiral subtheory) is not characterized by its modular invariant partition function. People tell me such an example exists, but noone could point out a concrete example or reference to me.

In the framework of Fuchs, Runkel, Schweigert etc. full CFTs are classified by Morita equivalence classes of Frobenius algebras in a modular tensor category $\mathcal C$ (the representation category of the chiral CFT), so my question more concretely:

I am looking for two special symmetric haploid Frobenius algebra objects $A,A'$ in a modular tensor category $\mathcal C$, which are not Morita equivalent but whose full centres $Z(A),Z(A')$ are equivalent as objects (they cannot be equivalent as algebra objects due to a result of Kong and Runkel), i.e. have the same modular invariant partition function $Z(A)_{ij}=Z(A')_{ij}$.

I am also fine with an example in any other formulation of CFT any reference in physics literature, non-degenerate braided subfactors or similar...