A star enshrouded in a Dyson sphere with high covering fraction may manifest itself as an optically subluminous object with a spectrophotometric distance estimate significantly in excess of its parallax distance. Using this criterion, the Gaia mission will in coming years allow for Dyson-sphere searches that are complementary to searches based on waste-heat signatures at infrared wavelengths. A limited search of this type is also possible at the current time, by combining Gaia parallax distances with spectrophotometric distances from ground-based surveys. Here, we discuss the merits and shortcomings of this technique and carry out a limited search for Dyson-sphere candidates in the sample of stars common to Gaia Data Release 1 and RAVE Data Release 5. We find that a small fraction of stars indeed display distance discrepancies of the type expected for nearly complete Dyson spheres. To shed light on the properties of objects in this outlier population, we present follow-up high-resolution spectroscopy for one of these stars, the late F-type dwarf TYC 6111-1162-1. The spectrophotometric distance of this object is about twice that derived from its Gaia parallax, and there is no detectable infrared excess. While our analysis largely confirms the stellar parameters and the spectrophotometric distance inferred by RAVE, a plausible explanation for the discrepant distance estimates of this object is that the astrometric solution has been compromised by an unseen binary companion, possibly a rather massive white dwarf ($\approx 1\ M_\odot$). This scenario can be further tested through upcoming Gaia data releases.