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. Does ${\rm SU}(N)$ have a pseudo-real (not real) representation? If so, how can we construct them explicitly?
Answer. It does if and only if $N=4q+2$. For $N=4q+2$, we 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 $$\rho(\lambda)=\Lambda^{2q+1}({\mathbb{C}}^N),$$ 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 $$(x,y)\mapsto x\wedge y\subset \Lambda^N ({\mathbb{C}}^N)={\mathbb{C}}.$$ Therefore, this representation is symplectic, and hence, pseudo-real, but not real. This example generalizes the standard representation of ${\rm SU}(2)$ in ${\mathbb{C}}^2$.