  We consider the $XY$ model with ferromagnetic (FM) and antinematic (AN)
nearest-neighbor interactions on a square lattice for a varying interaction
strength ratio. Besides the expected FM and AN quasi-long-range order (QLRO)
phases we identify at low temperatures another peculiar canted ferromagnetic
(CFM) QLRO phase, resulting from the competition between the collinear FM and
non-collinear AN ordering tendencies. In the CFM phase neighboring spins that
belong to different sublattices are canted by a non-universal (dependent on the
interaction strength ratio) angle and the ordering is characterized by a
fast-decaying power-law intra-sublattice correlation function. Compared to the
FM phase, in the CFM phase correlations are significantly diminished by the
presence of zero-energy domain walls due to the inherent degeneracy caused by
the AN interactions. We present the phase diagram as a function of the
interaction strength ratio and discuss the character of the respective phases
as well as the transitions between them.
