  It is generally taken for granted that our Universe is free of antimatter
objects and domains. This certitude has recently been challenged by the
possible detection of anti-helium nuclei by AMS-02. Should the observation be
confirmed, the existence of nearby antistars would make a plausible hypothesis
to explain the origin of the antinuclei. In this paper we use the 10-years
Fermi Large Area Telescope (LAT) gamma-ray source catalog to set constraints on
the abundance of antistars around the Sun. We identify in the catalog 14
antistar candidates not associated with any objects belonging to established
gamma-ray source classes and with a spectrum compatible with baryon-antibaryon
annihilation. We use them along with an estimate of the LAT sensitivity to
antistars to set upper limits on the local antistar fraction $f_{\bar{\ast}}$
with respect to normal stars. We provide parametric limits as a function of the
closest antistar mass, velocity, and surrounding matter density. We also employ
a novel Monte~Carlo method to set limits for a few hypotheses about the
antistar population. For a population with properties equivalent to those of
regular stars concentrated in the Galactic disk we obtain $f_{\bar{\ast}} < 2.5
\times 10^{-6}$ at 95\% confidence level, which is 20 times more constraining
than limits previously available. For a primordial population of antistars
distributed in the Galactic halo we obtain new local upper limits which
decrease as a function of antistar mass $M$ from $f_{\bar{\ast}} < 0.2$ at 95\%
confidence level for $M = 1 \; M_\odot$ to $f_{\bar{\ast}} < 1.6 \times
10^{-4}$ at 95\% confidence level for $M = 10 \; M_\odot$. By combining these
limits with existing microlensing constraints for lighter objects in the
Magellanic clouds, we infer that a primordial population of halo antistars must
have a density lower than $\mathcal{O}(10^{-5}\;\text{pc}^{-3})$ to
$\mathcal{O}(10^{-2}\;\text{pc}^{-3})$ depending on their masses. Our limits
can constrain models for the origin and propagation of antinuclei in cosmic
rays.
