  The exclusive reaction $\gamma p \to \bar K^0 K^+ n$ was studied in the
photon energy range between 1.6-3.8 GeV searching for evidence of the exotic
baryon $\Theta^+(1540)\to nK^+$. The decay to $nK^+$ requires the assignment of
strangeness $S=+1$ to any observed resonance. Data were collected with the CLAS
detector at the Thomas Jefferson National Accelerator Facility corresponding to
an integrated luminosity of 70 $pb^{-1}$. No evidence for the $\Theta^+$
pentaquark was found. Upper limits were set on the production cross section as
function of center-of-mass angle and $nK^+$ mass. The 95% CL upper limit on the
total cross section for a narrow resonance at 1540 MeV was found to be 0.8 nb.
