Resonance fluorescence in the Heitler regime provides access to single photons with coherence well beyond the Fourier transform limit of the transition, and holds the promise to circumvent environment-induced dephasing common to all solid-state systems. Here we demonstrate that the coherently generated single photons from a single self-assembled InAs quantum dot display mutual coherence with the excitation laser on a timescale exceeding 3 s. Exploiting this degree of mutual coherence, we synthesize near-arbitrary coherent photon waveforms by shaping the excitation laser field. In contrast to post-emission filtering, our technique avoids both photon loss and degradation of the single-photon nature for all synthesized waveforms. By engineering pulsed waveforms of single photons, we further demonstrate that separate photons generated coherently by the same laser field are fundamentally indistinguishable, lending themselves to the creation of distant entanglement through quantum interference.