Under the definition of Gennaro et al (link), a DKG protocol needs to satisfy “correctness” and “secrecy”. Correctness is divided into three sub-properties:
C1. All subsets of $t+1$ shares provided by honest players define the same unique secret key $x$.
C2. All honest parties have the same value of public key $y = g^x$ $mod$ $p$, where $x$ is the unique secret guaranteed by (C1).
C3. $x$ is uniformly distributed in $Z_q$ (and hence y is uniformly distributed in the subgroup generated by $g$).
Secrecy means that no information on $x$ can be learned by the adversary except for what is implied by the value $y = g^x$ $mod$ $p$. In terms of simulatability: for every (probabilistic polynomial-time) adversary $A$, there exists a (probabilistic polynomial-time) simulator $SIM$, such that on input an element $y$ in the subgroup of $Z_p$ generated by $g$, produces an output distribution which is polynomially indistinguishable from $A$'s view of a run of the DKG protocol that ends with $y$ as its public key output, and where $A$ corrupts up to $t$ parties.
They then continue and say: “The above is a minimal set of requirements needed in all known applications of such a protocol. In many applications a stronger version of (C1) is desirable, which reflects two additional aspects: (1) It requires the existence of an efficient procedure to build the secret $x$ out of $t+1$ shares; and (2) it requires this procedure to be robust, i.e. the reconstruction of $x$ should be possible also in the presence of malicious parties that try to foil the computation.”
We are looking for a DKG procedure that satisfies the first aspect of the stronger version of (C1), but not the second one. This means that in the presence of $t+1$ honest shares, reconstructing $x$ should be possible efficiently. However, in the presence of malicious parties, it is impossible to distinguish honest shares from malicious shares and reconstructing $x$ boils down to checking all the combinations of $t+1$ shares until one that comprises only of honest shares is found.