  Half-filled Landau levels admit the theoretically powerful fermion-vortex
duality but longstanding puzzles remain in their experimental realization as
$\nu_T=1$ quantum Hall bilayers, further complicated by Zheng et al's recent
numerical discovery of an unknown phase at intermediate layer spacing. Here we
propose that half-filled quantum Hall bilayers ($\nu_T=1$) at intermediate
values of the interlayer distance $d/\ell_B$ enter a phase with \textit{paired
exciton condensation}. This phase shows signatures analogous to the condensate
of interlayer excitons (electrons bound to opposite-layer holes) well-known for
small $d$ but importantly condenses only exciton pairs. To study it
theoretically we derive an effective Hamiltonian for bosonic excitons $b_k$ and
show that the single-boson condensate suddenly vanishes for $d$ above a
critical $d_{c1} \approx 0.95 l_B$. The nonzero condensation fraction
$n_0=\langle b(0) \rangle ^2$ at $d_{c1}$ suggests that the phase stiffness
remains nonzero for a range of $d>d_{c1}$ via an intermediate phase of
paired-exciton condensation, exhibiting $\langle bb \rangle \neq 0$ while
$\langle b \rangle =0$. Motivated by these results we derive a $K$-matrix
description of the paired exciton condensate's topological properties from
composite boson theory. The elementary charged excitation is a half meron with
$\frac{1}{4}$ charge and fractional self-statistics $\theta_s=\frac{\pi}{16}$.
Finally we argue for an equivalent description via the $d=\infty$ limit through
topological charge-$4e$ pairing of composite fermions. We suggest graphene
double layers should access this phase and propose various experimental
signatures, including an Ising transition $T_{Ising}$ below the
Berezinskii-Kosterlitz-Thouless transition $T_{BKT}$ at $d \sim d_{c1}$.
