  Establishing the fundamental relation between the homotopy invariants and the
band topology of Hamiltonians has played a critical role in the recent
development of topological phase research. In this work, we establish the
homotopy invariant and the related band topology of three-dimensional (3D)
real-valued Hamiltonians with two occupied and two unoccupied bands. Such a
real Hamiltonian generally appears in $\mathcal{PT}$ symmetric spinless fermion
systems where $\mathcal{P}$ and $\mathcal{T}$ indicate the inversion and
time-reversal symmetries, respectively. We show that the 3D band topology of
the system is characterized by two independent Hopf invariants when the
lower-dimensional band topology is trivial. Thus, the corresponding 3D band
insulator with nonzero Hopf invariants can be called a real Hopf insulator
(RHI). In sharp contrast to all the other topological insulators discovered up
to now, the topological invariants of RHI can be defined only when the fixed
number of both the occupied and unoccupied states are simultaneously
considered. Thus, the RHI belongs to the category of delicate topological
insulators proposed recently. We show that finite-size systems with slab
geometry support surface states with nonzero Chern numbers in a
$\mathcal{PT}$-symmetric manner independent of the Fermi level position, and
establish the bulk-boundary correspondence. We also discuss the bulk-boundary
correspondence of rotation symmetric RHIs using the returning Thouless pump.
