We present a noninvasive method for quantitative phase imaging through dynamically scattering media. A complex amplitude object, illuminated with coherent light, is captured through a dynamically scattering medium and a variable coded aperture, without the need for interferometric measurements or imaging optics. The complex amplitude of the object is computationally retrieved from intensity images that use multiple coded aperture patterns, employing a stochastic gradient descent algorithm. We demonstrate the proposed method both numerically and experimentally.