Let $G$ be a compact (anisotropic) real algebraic group.
Let $\rho\colon G\to {\rm GL}(n,{\mathbb{C}})$ be a complex linear (polynomial) representation of $G$.
Following OP, we say that $\rho$ is pseudo-real if $\bar \rho$ is isomorphic to $\rho$,
where $\bar \rho$ is the complex conjugate representation (given by complex conjugate matrices).
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
$$g\mapsto\rho(g^{-1})^T.$$
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 $B_-\,$.
Now assume that $\rho$ is irreducible and pseudo-real. Then $\rho$ preserves $B_+$ and $B_-\,$. However, by Schur's lemma $\rho$ may preserve, up to scalar, only one bilinear form. Thus either $B_-$=0 or $B_+=0$. We say that our pseudo-real representation is orthogonal or symplectic, respectively.
Assume $\rho$ is "real", that is, realizable over ${\mathbb{R}}$. This means that $\rho=\rho_0\otimes_{\mathbb{R}}{\mathbb{C}}$, where $\rho_0$ is a representation
in a real vector space $V_0$, and $V=V_0\otimes_{\mathbb{R}}{\mathbb{C}}$.
Since $G$ is compact, $V_0$ admits an invariant positive definite symmetric bilinear form $B_0$. Thus $\rho_0$ is an orthogonal representation, and so is $\rho$.
Conversely, assume that $\rho$ is ortogonal, that is, it preserves a non-degenerate symmetric bilinear form $B$.
On the other hand, since $G$ is compact, $\rho$ preserves a positive definite hermitian form $H$.
Comparing $B$ and $H$, we obtain that $\rho$ preserves a real structure on $V$, that is, antilinear involution. Thus $\rho$ is realizable over ${\mathbb{R}}$. For details see Serre, "Linear representations of finite groups", Section 13.2, Theorem 31 (Frobenius-Schur).
We conclude that an irreducible representations $\rho$ of a compact linear algebraic group $G$ is "real" is and only if it is orthogonal,
and $\rho$ is pseudo-real but not real if and only if it is symplectic.
Now assume that our compact linear algebraic group $G$ is semisimple.
Let $\rho$ be an irreducible complex representation of $G$ with highest weight $\lambda$,
given by its numerical labels $\lambda_i$ in the vertices $\alpha_i$ of the Dynkin diagram of $G$.
Hear $\lambda_i$ can be any natural numbers.
Exercises 9--13 in Section 4.3 in the book: Onishchik and Vinberg, "Lie Groups and Algebraic Groups", Berlin, Springer, 1990,
describe the orthogonal representations and the symplectic representations in terms of the numerical labels $\lambda_i$.
In particular, let $G={\rm SU}(N)$ with Dynkin diagram $A_{N-1}$.
Then symplectic irreducible representations (pseudo-real but not real irreducible representations) exist
if and only if $N=4q+2$ for some natural number $q$.
These are the representations $\rho(\lambda)$ for which:
(1) the numeric labels $\lambda_i$ are symmetric with the respect to the nontrivial automorphism of the Dynkin diagram $A_{4q+1}$,
and (2) $\lambda_{2q+1}$ is odd.
Originally OP asked the following question:
Question 1. Does $G={\rm SU}(N)$ have pseudo-real (not real) representations? If so, how can we construct them explicitly?
Answer 1. $G$ has pseudo-real non-real irreducible polynomial representations if and only if $N=4q+2$. 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
$$\rho(\lambda)=\Lambda^{N/2}({\mathbb{C}}^N),$$
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
$$(x,y)\mapsto x\wedge y\in \Lambda^N ({\mathbb{C}}^N)={\mathbb{C}}.$$
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.
