Hi JK,
we recently worked on a crystal with similar symmetry and I would recommend you to read the two papers listed below. If you visually do see the "weak" reflections, the space group is primitive orthorhombic (and not body-centered orthorhombic). The strong Patterson peak at ~0.5, ~0.5, ~0.5 results from pseudo-translational non-crystallographic symmetry (tNCS), as you correctly pointed out. As a consequence the intensity distribution is different compared to a "regular" crystal without tNCS, which affects molecular replacement and also coordinate refinement.
The tNCS could obscure the determination whether your crystallographic two-fold symmetry axes are screw axis or not, thus your space group could in principle also be P22121, P21212, P21221, P2122, P2212, P2221 or P222. This is nicely described in the first reference.
In light of the tNCS, you may also want to give Phaser a try for molecular replacement, I would try all eight primitive orthorhombic space groups (keyword: SGALTERNATIVE SELECT ALL).
Twinning could in theory be possible (e.g. a P2 or P21 twin with the unique beta cell angle very close to 90 degree [could be any of the three angles]), but difficult to detect together with the tNCS.
Best, Simon
J. A. Sundlov, A. M. Gulick, Structure determination of the functional domain interaction of a chimeric nonribosomal peptide synthetase from a challenging crystal with noncrystallographic translational symmetry. Acta Crystallogr D Biol Crystallogr 69, 1482-1492 (2013)10.1107/S0907444913009372).
R. J. Read, P. D. Adams, A. J. McCoy, Intensity statistics in the presence of translational noncrystallographic symmetry. Acta Crystallogr D Biol Crystallogr 69, 176-183 (2013); published online EpubFeb (10.1107/S0907444912045374).