  Renormalization-group theory predicts that the XXZ antiferromagnet in a
magnetic field along the easy Z-axis has asymptotically either a tetracritical
phase-diagram or a triple point in the field-temperature plane. Neither
experiments nor Monte Carlo simulations procure such phase diagrams. Instead,
they find a bicritical phase-diagram. Here this discrepancy is resolved: after
generalizing a ubiquitous condition identifying the tetracritical point, we
employ new renormalization-group recursion relations near the isotropic fixed
point, exploiting group-theoretical considerations and using accurate exponents
at three dimensions. These show that the experiments and simulations results
can only be understood if their trajectories flow towards the
fluctuation-driven first order transition (and the associated triple point),
but reach this limit only for prohibitively large system sizes or correlation
lengths. In the crossover region one expects a bicritical phase diagram, as
indeed is observed. A similar scenario may explain puzzling discrepancies
between simulations and renormalization-group predictions for a variety of
other phase diagrams with competing order parameters.
