  Motivated by recent experiments on heavy fermion materials CeCu$_2$Si$_2$ and
UBe$_{13}$, we develop a framework to capture generic properties of multiband
superconductors with strong Pauli paramagnetic effect (PPE). In contrast to the
single band case, the upper critical field $H_{\rm c2}$ can remain second order
transition even for strong PPE cases. The expected first order transition is
hidden inside $H_{\rm c2}$ and becomes a crossover due to the interplay of
multibandness. The present theory based on full self-consistent solutions of
the microscopic Eilenberger theory explains several mysterious anomalies
associated with the crossover and the "empty" vortex core state which is
observed by recent STM experiment on CeCu$_2$Si$_2$.
