Non-Poissonian photon statistics are usually traced to nonlinear interactions, nonclassical states, or engineered reservoirs. Here we show that measurement-based feedback can be alternatively deployed to control the Poissonian character of the measurement statistics in an externally-driven linear cavity. Each photon detection triggers a weak displacement operation whose phase determines how the feedback interferes with the coherent cavity field, resulting in a significant cavity response. Full counting statistics and two-time photon correlations show that the feedback phase controls the sign and strength of the induced correlations, continuously tuning the emission statistics between super- and sub-Poissonian regimes. At generic phases, the Mandel parameter responds linearly to the feedback amplitude, whereas at specific phases, the leading response becomes quadratic. The same phase dependence is also found in higher-order cumulants of the emission statistics. These results establish measurement-based feedback as a route to controlling photon statistics in otherwise linear driven-dissipative systems.