As an example consider the complex $16$-dimensional representation of $\mathfrak{so}(10)$. When $\mathfrak{so}(10)$ is reduced to the subalgebra $\mathfrak{so}(9)$, the complex $16$-dimensional representation of $\mathfrak{so}(10)$ becomes the real $16$-dimensional representation of $\mathfrak{so}(9)$
$$ 16 \rightarrow 16 $$
Another example is the complex $27$ of $\mathfrak{e}_6$, which becomes the real $26\oplus 1$, when $\mathfrak{e}_6$ is reduced to $\mathfrak{f}_4$.
A complex $16$ has $16\cdot 2=32$ degrees of freedom, a real $16$ only $16$. Where do these degrees of freedom go? Surely they can't get lost so do we really get somehow
$$ 16 \rightarrow 16 \oplus i 16 $$
or something like that?
The other way round things are more transparent. When a algebra with only real representations becomes a subalgebra with complex representation, we always have
$$ R= r_1\oplus \bar r_1 \oplus \ldots$$