  Doped van der Waals heterostructures host layer-hybridized trions, i.e.
charged excitons with layer-delocalized constituents holding promise for highly
controllable optoelectronics. Combining a microscopic theory with
photoluminescence (PL) experiments, we demonstrate the electrical tunability of
the trion energy landscape in naturally stacked WSe$_2$ bilayers. We show that
an out-of-plane electric field modifies the energetic ordering of the lowest
lying trion states, which consist of layer-hybridized $\Lambda$-point electrons
and layer-localized K-point holes. At small fields, intralayer-like trions
yield distinct PL signatures in opposite doping regimes characterized by weak
Stark shifts in both cases. Above a doping-asymmetric critical field,
interlayer-like species are energetically favored and produce PL peaks with a
pronounced Stark red-shift and a counter-intuitively large intensity arising
from efficient phonon-assisted recombination. Our work presents an important
step forward in the microscopic understanding of layer-hybridized trions in van
der Waals heterostructures and paves the way towards optoelectronic
applications based on electrically controllable atomically-thin semiconductors.
