  In this paper we analyze Abelian-Higgs strings in a phenomenological model
that takes quantum effects in curved space-time into account. This model, first
introduced by Rastall, cannot be derived from an action principle. We formulate
phenomenological equations of motion under the guiding principle of minimal
possible deformation of the standard equations. We construct string solutions
that asymptote to a flat space-time with a deficit angle by solving the set of
coupled non-linear ordinary differential equations numerically. Decreasing the
Rastall parameter from its Einstein gravity value we find that the deficit
angle of the space-time increases and becomes equal to $2\pi$ at some critical
value of this parameter that depends on the remaining couplings in the model.
For smaller values the resulting solutions are supermassive string solutions
possessing a singularity at a finite distance from the string core. Assuming
the Higgs boson mass to be on the order of the gauge boson mass we find that
also in Rastall gravity this happens only when the symmetry breaking scale is
on the order of the Planck mass. We also observe that for specific values of
the parameters in the model the energy per unit length becomes proportional to
the winding number, i.e. the degree of the map $S^1 \rightarrow S^1$. Unlike in
the BPS limit in Einstein gravity, this is, however, not connect to an
underlying mathematical structure, but rather constitutes a would-be-BPS bound.
