Since $G$ is compact, $\rho$ preserves a positive definite hermitian form $H$ on ${\mathbb{C}}^n$.
After changing the basis, we may assume that $H$ is given by the init matrix ${\rm diag}(1,\dots,1).$
Then
$$\bar\rho(g)=(\rho(g)^T)^{-1}=\rho(g^{-1})^T,$$
where $^T$ denotes the transpose matrix.
We write $V$ for ${\mathbb{C}}^n$.
We denote by $\rho^\vee$ the contragredient representation to $\rho$, that is, the natural representation of $G$ in the dual space $V^\vee$.
Then in a suitable basis it is given by
[g\mapsto\rho(g^{-1})^T.]$$g\mapsto\rho(g^{-1})^T.$$
We see that the representation $\rho$ in $V={\mathbb{C}}^n$ is pseudo-real if and only if
it is isomorphic to the contragredient representation $\rho^\vee$ in $V^\vee$.
It is easy to see that this holds if and only if $\rho$ preserves a non-degenerate bilinear form $B\colon V\times V\to {\mathbb{C}}$.
We can canonically write $B$ as a sum $B=B_+ +B_-$, where $B_+$ is symmetric and $B_-$ is alternating.
Then $\rho$ must preserve both $B_+$ and $B_-\,$.
Question 1. Does ${\rm SU}(N)$$G={\rm SU}(N)$ have a pseudo-real (not real) representationrepresentations? If so, how can we construct them explicitly?
Answer 1. It does $G$ has pseudo-real non-real irreducible polynomial representations if and only if $N=4q+2$.
For For $N=4q+2$, we can take $\lambda_i=0$ for $i\neq 2q+1$, and $\lambda_{2q+1}=1$.
According to tables in the book of Onishchik and Vinberg (Table 5 and Table 1), then
$$\rho(\lambda)=\Lambda^{2q+1}({\mathbb{C}}^N),$$$$\rho(\lambda)=\Lambda^{N/2}({\mathbb{C}}^N),$$
where we write ${\mathbb{C}}^N$ for the standard representation of ${\rm SU}(N)$ in ${\mathbb{C}}^N$.
It is well known that this representation is irreducible.
It preserves the alternating bilinear form
$$(x,y)\mapsto x\wedge y\subset \Lambda^N ({\mathbb{C}}^N)={\mathbb{C}}.$$$$(x,y)\mapsto x\wedge y\in \Lambda^N ({\mathbb{C}}^N)={\mathbb{C}}.$$
Therefore, this representation is symplectic, and hence, pseudo-real, but not real.
This example generalizes the standard representation of ${\rm SU}(2)$ in ${\mathbb{C}}^2$.
ADDENDUM. Slightly generalizing, we may consider the following more general question:
Question 2. Consider the not necessary compact real algebraic group
$G={\rm SU}(N-s,s)$. Does $G$ have pseudo-real non-real representations?
Answer 2. $G$ has pseudo-real non-real irreducible finite-dimensional polynomial representations if and only if $N=2m$ and $m-s$ is odd. As an example of a pseudo-real non-real representation we can again take $\Lambda^{N/2}({\mathbb{C}}^N)$.
In particular, if $s=0$, then $G={\rm SU}(N)$ has pseudo-real non-real representations if and only if $N=2m$ and $m$ is odd. Thus Answer 2 is compatible with Answer 1.
A proof for Answer 2 can be obtained by combining results of the paper by Tits Représentations linéaires irréductibles d'un groupe réductif sur un corps quelconque and tables of Tits invariants over $\mathbb{R}$ in my appendix to
the preprint by Lucy Moser-Jauslin and Ronan Terpereau Real structures on horospherical varieties.
According to these tables, the Tits invariant of ${\rm SU}(N-s,s)$ is nontrivial if and only if $N=2m$ and $m-s$ is odd.
