  We investigate the time evolution of spherically symmetric boson stars in
Palatini $f(\mathcal{R})$ gravity through Numerical Relativity computations.
Employing a novel approach that establishes a correspondence between modified
gravity with scalar matter and General Relativity with modified scalar matter,
we are able to use the techniques of Numerical Relativity to simulate these
systems. Specifically, we focus on the quadratic theory
$f(\mathcal{R})=\mathcal{R}+\xi\mathcal{R}^2$ and compare the obtained
solutions with those in General Relativity, exploring both positive and
negative values of the coupling parameter $\xi$. Our findings reveal that boson
stars in Palatini $f(\mathcal{R})$ gravity exhibit both stable and unstable
evolutions. The latter give rise to three distinct scenarios: migration towards
a stable configuration, complete dispersion, and gravitational collapse leading
to the formation of a baby universe structure.
