  For a congruence level $H \subset \mathrm{GL}_2(\hat{\mathbb{Z}})$, the
formation of the modular curve $X_H$, i.e., of the coarse moduli space of the
level $H$ modular stack $\mathscr{X}_H$, is known to commute with arbitrary
base change in a wide range of cases. We exhibit infinitely many $H$, for
instance, $H = \Gamma_1(4)$, for which this coarse base change property fails.
In our examples failure is witnessed for base change to $\mathbb{F}_2$ and for
any $\mathbb{Z}_{(2)}$-fiberwise dense open substack of $\mathscr{X}_H$. These
examples fill in several open entries in a table in the book of Katz and Mazur.
