We formulate the "real integral Hodge conjecture", a version of the integral
Hodge conjecture for real varieties, and raise the question of its validity for
cycles of dimension 1 on uniruled and Calabi-Yau threefolds and on rationally
connected varieties. We relate it to the problem of determining the image of
the Borel-Haefliger cycle class map for 1-cycles, with the problem of deciding
whether a real variety with no real point contains a curve of even geometric
genus and with the problem of computing the torsion of the Chow group of
1-cycles of real threefolds. New results about these problems are obtained
along the way.Comment: 67 pages; v2: minor modifications; v3: Section 1.1.3 slightly
expanded, final versio