We observed the field of the Fermi source 3FGL J0838.8−2829 in optical and X-rays, initially motivated by the cataclysmic variable (CV) 1RXS J083842.1−282723 that lies within its error circle. Several X-ray sources first classified as CVs have turned out to be γ-ray emitting millisecond pulsars (MSPs). We find that 1RXS J083842.1−282723 is in fact an unusual CV, a stream-fed asynchronous polar in which accretion switches between magnetic poles (that are ≈ 120• apart) when the accretion rate is at minimum. High-amplitude X-ray modulation at periods of 94.8±0.4 minutes and 14.7±1.2 hr are seen. The former appears to be the spin period, while the latter is inferred to be one-third of the beat period between the spin and the orbit, implying an orbital period of 98.3 ± 0.5 minutes. We also measure an optical emission-line spectroscopic period of 98.413 ± 0.004 minutes, which is consistent with the orbital period inferred from the X-rays. In any case, this system is unlikely to be the γ-ray source. Instead, we find a fainter variable X-ray and optical source, XMMU J083850.38−282756.8, that is modulated on a time scale of hours in addition to exhibiting occasional sharp flares. It resembles the black widow or redback pulsars that have been discovered as counterparts of Fermi sources, with the optical modulation due to heating of the photosphere of a low-mass companion star by, in this case, an as-yet undetected MSP. We propose XMMU J083850.38−282756.8 as the MSP counterpart of 3FGL J0838.8−2829.