As alluded to in the comments, this is just an error of terminology, and the terminology is indeed unfortunate. Here's what's happening:
$\mathfrak{so}(10)$ has two inequivalent irreps on $16$-dimensional complex vector spaces; they are the ones with highest weight vector $\varpi_4$ and $\varpi_5$ respectively, where $\varpi_4$ and $\varpi_5$ are the two fundamental weights which correspond to (i.e. are dual to the simple roots which form) the two "horns" in the Dynkin diagram $D_5$. By the way, they seem to be prominent. They feature in https://mathoverflow.net/q/40564/27465 and recently their lifts to the group level came up in Embed a Spin group to a special unitary group.
These two irreps are, as said, not equivalent to each other, but they happen to be conjugates of each other.
On the other hand, $\mathfrak{so}(9)$ has up to equivalence just one irrep on a $16$-dimensional complex vector space; it is the one whose highest weight vector is $\varpi_4$, which corresponds to the short root in the Dynkin diagram $B_4$.
This representation is self-conjugate (like all representations of something of type $B$).
Now, the branching you cite implies that if you restrict either of those two $16$-dimensional irreps of $\mathfrak{so}(10)$ to $\mathfrak{so}(9)$, you get that $16$-dimensional irrep $V_{\mathfrak{so}(9)}(\varpi_4)$.
What confuses you and others is that some people (especially physicists) would call those two irreps of $\mathfrak{so}(10)$ "complex", and the irrep of $\mathfrak{so}(9)$ "real". However, that $\mathfrak{so}(9)$-representation is still on a $16$-dimensional complex vector space. The terminology "real" here means that this one has a "real structure": when you view that $\mathbb C^{16}$ as a $32$-dimensional real vector space, you can find a $16$-dimensional real subspace which is invariant under the $\mathfrak{so}(9)$-action. (Also and equivalently, this complex representation comes from a real representation by complexifying this $16$-dimensional real subspace, as Vincent suggests in a comment.) Whereas no such real subspace will be invariant under either of the $\mathfrak{so}(10)$-actions; which happens to follow from the fact stated above that they are conjugate but inequivalent; which in this terminology means those irreps are "truly complex". But, stressed again, all these irreps are on $\mathbb C^{16}$, and no dimension is "lost" in the restriction / branching.
Maybe after asking this question you noticed that bad terminology, because it was the content of a question you asked one day later: Are the physics and math definitions of a complex representation equivalent? A related confusion came up in What property of the root system means a Lie algebra has complex structure?
It might be worthwhile to remark that the above situation repeats itself almost verbatim for every inclusion $\mathfrak{so}(4n+1) \subset \mathfrak{so}(4n+2)$, where the bigger Lie algebra has two conjugate "truly complex" irreps of dimension $2^{2n}$ (which are minuscule and apparently called "half-spin"), and both of them "branch" to just one complex irrep, of same dimension, of the smaller group (also minuscule and apparently called "spin"), and the latter one has a real structure.
One can see that explicitly in the case $n=1$, where the inclusion $\mathfrak{so}(5) \subset \mathfrak{so}(6)$ translates via standard isomorphisms to
$\mathfrak{sp}(2) \subset \mathfrak{su}(4)$ (where $\mathfrak{sp}(2)$ denotes the compact real form of the symplectic Lie algebra of type $C_2$), which one can both write as certain complex $4\times 4$-matrices, and then one can check that the natural action of the bigger one on $\mathbb C^4$ is non-equivalent to its conjugate, whereas the action of $\mathfrak{sp}(2)$ is equivalent to its conjugate and has a "real structure".
