A generalization of periodic clustered-dot halftones is proposed, wherein the phase of the halftone spots is modulated using a secondary signal. The process is accomplished by using an analytic halftone threshold function that allows halftones to be generated with controlled phase variation in different regions of the printed page. The method can also be used to modulate the screen frequency, albeit with additional constraints. Visible artifacts are minimized/eliminated by ensuring the continuity of the modulation in phase. Limitations and capabilities of the method are analyzed through a quantitative model. The technique can be exploited for two applications that are presented in this paper: a) embedding watermarks in the halftone image by encoding information in phase or in frequency and b) modulating the screen frequency according to the frequency content of the continuous tone image in order to improve spatial and tonal rendering. Experimental performance is demonstrated for both applications.