  Results are presented from numerical simulations of the
Einstein-Maxwell-Higgs equations with a broken U(1) symmetry. Coherent
nontopological soliton solutions are shown to exist that separate an Anti-de
Sitter (AdS) true vacuum interior from a Reissner-Nordstrom (RN) false vacuum
exterior. The stability of these bubble solutions is tested by perturbing the
charge of the coherent solution and evolving the time-dependent equations of
motion. In the weak gravitational limit, the short-term stability depends on
the sign of $(\omega/ Q) \, \partial_\omega Q$, similar to q-balls. The
long-term end state of the perturbed solutions demonstrates a rich structure
and is visualized using "phase diagrams." Regions of both stability and
instability are shown to exist for $\kappa_g \lesssim 0.015$, while solutions
with $\kappa_g \gtrsim 0.015$ were observed to be entirely unstable. Threshold
solutions are shown to demonstrate time-scaling laws, and the space separating
true and false vacuum end states is shown to be fractal in nature, similar to
oscillons. Coherent states with superextremal charge-to-mass ratios are shown
to exist and observed to collapse or expand, depending on the sign of the
charge perturbation. Expanding superextremal bubbles induce a phase transition
to the true AdS vacuum, while collapsing superextremal bubbles can form
nonsingular strongly gravitating solutions with superextremal RN exteriors.
