Assume that $(M, [\lambda, \mu])$ defines an embeddable 3 dimensional CR structure where $\lambda$ is a real form and $\mu$ is a complex 1-form. Because $M$ is embeddable, $\mu=dz$ for some complex variable $z$.
We also assume that the equation $\partial_z \phi=f$ has a complex solution $\phi$ where $f$ is a non zero complex function. Can this equation also have any real solution?
If I understand the question correctly, the answer in general is No.
It is not clear from the question, what is the exact relationship between the CR structure and the form $\mu$, as well as the meaning of $\partial_z$ (for a partial derivative to be defined, you would need a full coordinate chart).
However, on any (almost) CR structure, there is the canonical $\partial_b$ operator sending a complex function $\phi$ to a $(1,0)$ form $f$ on the complex tangent bundle $T^cM := TM\cap JTM$, where $J$ is the complex structure of the complex manifold where $M$ is embedded. Note that $\partial_b\phi$ is only invariantly defined as form on the complex tangent space of $M$, not the full tangent space.
Now, restating the question for the $\partial_b$ operator, the easiest case to consider is that of the Levi-flat $M\subset \mathbb C^2_{z,w}$ given by $\Im w =0$. That effectively reduces $\partial_b$ to the one-dimensional $\partial$ operator along the complex tangent direction, i.e. in $z$-variable. Then $\partial \phi = f$ is always locally solvable for $f$ smooth by means of the Cauchy kernel or, even simpler, for $f$ real-analytic by separating monomials.
On the other hand, a real solution $\phi$ would require certain symmetries from $f$ with respect to the complex conjugation. The simplest of these symmetries appears at the lowest order level, for the complex Hessian. If $\phi$ is real, its complex Hessian $-i\bar\partial \partial \phi$ is invariant under conjugation. On the other hand, $i\bar\partial f = i\bar\partial \partial \phi$ does not have any conjugation invariance for general $f$.
