  The $\gamma p \to K^+ \pi^0 \Lambda$ and $\gamma p \to K^+ \pi \Sigma $
reactions are studied in the kinematic region where the $\pi^0 \Lambda$(1116)
and $\pi\Sigma$(1192) pairs originate dominantly from the decay of the
$\Sigma$(1385) and $\Lambda$(1405) resonances. We consider laboratory photon
energies around 2 GeV, significantly above the threshold for producing the $K^+
\Sigma$(1385) and $K^+ \Lambda$(1405) final states. We compute for both
reactions the process in which the ingoing photon dissociates into a real K$^+$
and a virtual K$^-$, the off-shell K$^-$ scattering subsequently off the proton
target to produce the $\pi^0 \Lambda$ or $\pi \Sigma$ pair. The $K^- p \to
\pi^0 \Lambda$ and $K^- p \to \pi \Sigma$ amplitudes are calculated in the
framework of a chiral coupled-channel effective field theory of meson-baryon
scattering. The structure of the amplitudes reflects the dominance of the
$\Lambda(1405)$ in the $\pi \Sigma$ channel and of the $\Sigma(1385)$ in the
$\pi \Lambda$ channel. The full pion-hyperon final state interaction is
included in these amplitudes. We extract from the calculated cross section the
gauge-invariant double kaon pole term. We found this term to be large and
leading to sizeable cross sections for both the $\gamma p \to K^+ \pi^0
\Lambda$ and $\gamma p \to K^+ \pi \Sigma $ reactions, in qualitative agreement
with the scarce data presently available. Accurate measurements of these cross
sections should make it possible to extract the contribution of the double kaon
pole and hence to assess the possibility of studying kaon-nucleon dynamics ust
below threshold through these reactions.
