  The theory of elliptic modular forms has gained significant momentum from the
discovery of relaxed yet well-behaved notions of modularity, such as mock
modular forms, higher order modular forms, and iterated Eichler-Shimura
integrals. Applications beyond number theory range from combinatorics,
geometry, and representation theory to string theory and conformal field
theory. We unify these relaxed notions in the framework of vector-valued
modular forms by introducing a new class of
$\mathrm{SL}_{2}(\mathbb{Z})$-representations: virtually real-arithmetic types.
The key point of the paper is that virtually real-arithmetic types are in
general not completely reducible. We obtain a rationality result for Fourier
and Taylor coefficients of associated modular forms.
