The answer is 'yes' there do exist such functions that are non-constant with singularities only along surfaces $\Sigma\subset\mathbb{C}^2$, and here is how one can understand them:

First, it helps to change coordinates, though, perhaps, a little more subtly than Fedor Petrov suggested: Let
$$
y_1 = \tfrac i2({\overline z}_1+{\overline z}_2 - z_1 + z_2)
\quad\text{and}\quad
y_2 = \tfrac12({\overline z}_1+{\overline z}_2 + z_1 - z_2)
$$
Then $(y_1,y_2):\mathbb{C}^2\to\mathbb{C}^2$ is a diffeomorphism. Using the complex structure on $\mathbb{C}^2$ for which $y_1$ and $y_2$ are holomorphic coordinates, we find that
$$
\frac{\partial f}{\partial{\overline y}_1} = \frac{\partial f}{\partial{\overline y}_2} = 0,
$$
so that $f$ is a holomorphic function of $y_1$ and $y_2\,$. Moreover, we have the periodicity relations
$$
f(y_1{+}1,y_2)= f(y_1,y_2{+}1) = f(y_1,y_2)
$$
while
$$
f(y_1{+}i,y_2)= \mathrm{e}^{ik_1}f(y_1,y_2)
\quad\text{and}\quad
f(y_1,y_2{+}i) = \mathrm{e}^{-ik_2}f(y_1,y_2)
$$

Now, one can construct the general meromorphic $f$ that satisfies these condition as follows:

First, let $p_1$ and $p_2$ be (not-identically-vanishing) meromorphic functions on $\mathbb{C}$ that satisfy the period relations
$$
p_1(y+1) = p_1(y)\quad\text{and}\quad
p_1(y+i) = \mathrm{e}^{ik_1}\,p_1(y)
$$
and
$$
p_2(y+1) = p_2(y)\quad\text{and}\quad
p_2(y+i) = \mathrm{e}^{-ik_2}\,p_2(y).
$$
Techniques for constructing meromorphic functions on $\mathbb{C}$ satisfying such double-periodicity relations are well-known, using $\vartheta$-series or the theory of elliptic curves. The point is that, if $C = \mathbb{C}/\mathbb{Z}[i]$ is the square torus (which is an elliptic curve), then the above relations on meromorphic functions on $\mathbb{C}$ essentially describe the meromorphic sections of two flat complex line bundles $E_1$ and $E_2$ over $C$, i.e., elements of $\mathrm{Pic}_0(C)$.

Given $p_1$ and $p_2$, then the general meromorphic $f$ that satisfies the above conditions can be written as a product
$$
f(y_1,y_2) = p_0(y_1,y_2)\,p_1(y_1)\,p_2(y_2),
$$
where $p_0$ comes from any meromorphic function on $C\times C$, i.e., $p_0$ is a meromorphic function on $\mathbb{C}^2$ that satisfies
$$
p_0(y_1{+}m,y_2)= p_0(y_1,y_2{+}m) = p_0(y_1,y_2)
$$
for any Gaussian integer $m\in\mathbb{Z}[i]$.

Note that the 'singularities' of $f$ occur along surfaces in $\mathbb{C}^2$ that are complex curves in the $y$-coordinates whose images in $C\times C$ are algebraic curves.

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