We demonstrate the operation of a quadrature phase-shifted extrinsic Fabry-Perot fiber-optic sensor for the detection of the amplitude and the relative polarity of dynamically varying strain. Two laterally displaced single-mode fibers inserted within a hollow silica tube form the 90 degrees phase-shifted sensing system. A multimode fiber, placed in the tube facing the two fibers, acts as a reflector, thereby creating an air gap that acts as a Fabry-Perot cavity. A theoretical description of the sensor is given, and its operation as a dynamically varying strain sensor is described. Strain sensitivities of 5.54 degrees phase shift/microstrain cm(-1) are obtained.