A vector field can be written in terms of irrotational and a divergence-free components. Using a 2D velocity field as an example,
$ \vec v = -\nabla \phi + \nabla \times \vec \Psi$
Where $\vec \Psi$ is a vector potential, which in fluid mechanics is only guaranteed to exist if we're working in two dimensions so that $\vec \Psi = (0,0,\psi)$, where $\psi$ is called the stream function.
There are many sources I can find that say that an incompressible flow ($\nabla \cdot \vec v = 0$) simplifies to $ \vec v = \nabla \times \vec \Psi$ Here is one such example, although the Wikipedia article on stream functions implies the same.
This seems incorrect to me, since taking the divergence of both sides of this equation $ \nabla \cdot \vec v = -\nabla \cdot \nabla \phi + \nabla \cdot \nabla \times \vec \Psi$ simply yields the Laplace equation, $\nabla^2 \phi = 0$. This means that as long as $\phi$ is a nonzero harmonic function, I can have a velocity field in an incompressible fluid that has an irrotational component. Is there an additional constraint that forces $\nabla \phi = 0$ in order for $\nabla \cdot \vec v =0$?