  We present a comprehensive experimental and theoretical study of the
pyrochlore Tb$_2$Ge$_2$O$_7$, an exemplary realization of a material whose
properties are dominated by competition between magnetic dipolar and electric
quadrupolar correlations. The dipolar and quadrupolar correlations evolve over
three distinct regimes that we characterize via heat capacity, elastic and
inelastic neutron scattering. In the first regime, above $T^*=1.1$ K,
significant quadrupolar correlations lead to an intense inelastic mode that
cannot be accounted for within a scenario with solely magnetic dipole-dipole
correlations. The onset of extended dipole correlations occurs in the
intermediate regime, between $T^*=1.1$ K and $T_c = 0.25$ K, with the formation
of a collective paramagnetic state characterized by extended ferromagnetic
canted spin ice domains. Here, long-range order is impeded not only by the
usual frustration operating in classical spin ice systems, but also by a
competition between dipolar and quadrupolar correlations. Finally, in the
lowest temperature regime, below $T_c=0.25$ K, there is an abrupt and
significant increase in the dipole ordered moment. The majority of the ordered
moment remains tied up in the ferromagnetic spin ice-like state, but an
additional $\mathbf{k}=(0,0,1)$ antiferromagnetic order parameter also
develops. Simultaneously, the spectral weight of the inelastic mode, which is a
proxy for the quadrupolar correlations, is observed to drop, indicating that
dipole order ultimately wins out. Tb$_2$Ge$_2$O$_7$ is therefore a remarkable
platform to study intertwined dipolar and quadrupolar correlations in a
magnetically frustrated system and provides important insights into the physics
of the whole family of terbium pyrochlores.
