  An identifying code $C$ of a graph $G$ is a dominating set of $G$ such that
any two distinct vertices of $G$ have distinct closed neighbourhoods within
$C$. The smallest size of an identifying code of $G$ is denoted
$\gamma^{\text{ID}}(G)$. When every vertex of $G$ also has a neighbour in $C$,
it is said to be a total dominating identifying code of $G$, and the smallest
size of a total dominating identifying code of $G$ is denoted by
$\gamma_t^{\text{ID}}(G)$.
  Extending similar characterizations for identifying codes from the
literature, we characterize those graphs $G$ of order $n$ with
$\gamma_t^{\text{ID}}(G)=n$ (the only such connected graph is $P_3$) and
$\gamma_t^{\text{ID}}(G)=n-1$ (such graphs either satisfy
$\gamma^{\text{ID}}(G)=n-1$ or are built from certain such graphs by adding a
set of universal vertices, to each of which a private leaf is attached).
  Then, using bounds from the literature, we remark that any (open and closed)
twin-free tree of order $n$ has a total dominating identifying code of size at
most $\frac{3n}{4}$. This bound is tight, and we characterize the trees
reaching it. Moreover, by a new proof, we show that this bound actually holds
for the larger class of all twin-free graphs of girth at least 5. The cycle
$C_8$ also attains this bound. We also provide a generalized bound for all
graphs of girth at least 5 (possibly with twins).
  Finally, we relate $\gamma_t^{\text{ID}}(G)$ to the related parameter
$\gamma^{\text{ID}}(G)$ as well as the location-domination number of $G$ and
its variants, providing bounds that are either tight or almost tight.
