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We present the SHEEP survey for serendipitously detected hard X-ray sources in ASCA GIS images. In a survey area of ∼40 deg², 69 sources were detected in the 5-10 keV band to a limiting flux of ∼10^–13 ergs.cm^–2.s^–1. The number counts agree with those obtained by the similar BeppoSAX High-Energy Large Area Survey (HELLAS), and both are in close agreement with ASCA and BeppoSAX 2-10 keV surveys. Spectral analysis of the SHEEP sample reveals that the 2-10 and 5-10 keV surveys do not sample the same populations, however, as we find considerably harder spectra, with an average Γ∼1.0, assuming no absorption. The implication is that the agreement in the number counts is coincidental, with the 5-10 keV surveys gaining approximately as many hard sources as they lose soft ones when compared to the 2-10 keV surveys. This is hard to reconcile with standard active galactic nucleus (AGN) ``population synthesis'' models for the X-ray background, which posit the existence of a large population of absorbed sources. We find no evidence of the population hardening at faint fluxes, with the exception that the few very brightest objects are anomalously soft. Fifty-three of the SHEEP sources have been covered by ROSAT in the pointed phase. Of these, 32 were detected. An additional three were detected in the RASS. As expected, the sources detected with ROSAT are systematically softer than those detected with ASCA alone and than the sample as a whole. Although they represent a biased subsample, the ROSAT positions allow relatively secure catalog identifications to be made. We find associations with a wide variety of AGNs and a few clusters and groups. At least two X-ray sources identified with high-z QSOs present very hard X-ray spectra indicative of absorption, despite the presence of broad optical lines. A possible explanation for this is that we are seeing relatively dust-free ``warm absorbers'' in high-luminosity/redshift objects. Color analysis indeed indicates that many of the spectra are not consistent with a simple, absorbed power law. The spectra are likely to be complex, with an absorbed hard power law and scattered or ``leaky'' component in the soft X-rays. Many are also consistent with a reflection-dominated spectrum. Our analysis defines a new, hard X-ray-selected sample of objects–mostly AGNs–that is less prone to bias due to obscuration than previous optical or soft X-ray samples. They are therefore more representative of the population of AGNs in the universe in general, and the SHEEP survey should produce bright examples of the sources that make up the hard X-ray background, the majority of which have recently been resolved by Chandra. This should help elucidate the nature of the new populations.