Noteworthy strides continue to be made in the development of full-duplex millimeter wave (mmWave) communication systems, but most of this progress has been built on theoretical models and validated through simulation. In this work, we conduct a long overdue real-world evaluation of full-duplex mmWave systems using off-the-shelf 60 GHz phased arrays. Using an experimental full-duplex base station, we collect over 200,000 measurements of self-interference by electronically sweeping its transmit and receive beams across a dense spatial profile, shedding light on the effects of the environment, array positioning, and beam steering direction. We then call attention to five key challenges faced by practical full-duplex mmWave systems and, with these in mind, propose a general framework for beamforming-based full-duplex solutions. Guided by this framework, we introduce a novel solution called STEER+, a more robust version of recent work called STEER, and experimentally evaluate both in a real-world setting with actual downlink and uplink users. Rather than purely minimize self-interference as with STEER, STEER+ makes use of additional measurements to maximize spectral efficiency, which proves to make it much less sensitive to one's choice of design parameters. We experimentally show that STEER+ can reliably reduce self-interference to near or below the noise floor while maintaining high SNR on the downlink and uplink, thus enabling full-duplex operation purely via beamforming.