  We give an unconditional construction of K3 surfaces over finite fields with
given L-function, up to finite extensions of the base fields, under some mild
restrictions on the characteristic. Previously, such results were obtained by
Taelman assuming semistable reduction. The main contribution of this paper is
to make Taelman's proof unconditional. We use some results of Nikulin and
Bayer-Fluckiger to construct an appropriate complex projective K3 surface with
CM which admits an elliptic fibration with a section, or an ample line bundle
of low degree. Then using Saito's construction of strictly semistable models
and applying a slight refinement of Matsumoto's good reduction criterion for K3
surfaces, we obtain a desired K3 surface over a finite field.
