  We extend Allard's celebrated rectifiability theorem to the setting of
varifolds with locally bounded first variation with respect to an anisotropic
integrand. In particular, we identify a sufficient and necessary condition on
the integrand to obtain the rectifiability of every \(d\)-dimensional varifold
with locally bounded first variation and positive \(d\)-dimensional density. In
codimension one, this condition is shown to be equivalent to the strict
convexity of the integrand with respect to the tangent plane.
