The existence of exotic hadrons not allowed by a naive qqq and qq¯ quark model would provide an opportunity to study the strong reaction in rare form. The pentaquark state named Θ ⁺, with minimal constituent quark content uudds¯, has been predicted with a low mass of 1530 MeV/c² and width no larger than 15 MeV/c². Numerous experiments have since reported evidence of the Θ⁺ in both nK⁺ and pK ⁰ decay modes with the predicted characteristics, but the world's data is inconclusive due to limited statistics and background understanding. In this study, quasi-free photoproduction off a neutron in the reaction γ d → pp[special characters omitted]K^– is exclusively measured using the CLAS detector system at Thomas Jefferson National Accelerator Facility. Tagged photons with energies from 1.6 to 3.6 GeV are measured in concidence with pπ⁺π^– K^– in the spectrometer. A multidimensional model based on Breit-Wigner distributions for the known background resonances is fit to the data with an unbinned likelihood method. No significant evidence of Θ⁺ production is found. An upper limit on the integrated total cross section of γn → Θ ⁺K^– is estimated at 3.3 nb with 95% confidence level.