  By taking the nucleon-to-quark phase transition within a neutron star as an
example, we present a thermodynamically consistent method to calculate the
equation of state of ambient matter so that transitions that are intermediate
to those of the familiar Maxwell and Gibbs constructions can be described. This
method does not address the poorly known surface tension between the two phases
microscopically (as, for example, in the calculation of the core pasta phases
via the Wigner-Seitz approximation) but instead combines the local and global
charge neutrality conditions characteristic of the Maxwell and Gibbs
constructions, respectively. Overall charge neutrality is achieved by dividing
the leptons to those that obey local charge neutrality (Maxwell) and those that
maintain global charge neutrality (Gibbs). The equation of state is obtained by
using equilibrium constraints derived from minimizing the total energy density.
The results of this minimization are then used to calculate neutron star
mass-radius curves, tidal deformabilities, equilibrium and adiabatic sound
speeds, and nonradial $g$-mode oscillation frequencies for several intermediate
constructions. Various quantities of interest transform smoothly from their
Gibbs structures to those of Maxwell as the local-to-total electron ratio
$\eta$, introduced to mimic the hadron-to-quark interface tension from $0$
(Gibbs) to $\infty$ (Maxwell), is raised from $0$ to $1$. A notable exception
is the $g$-mode frequency for the specific case of $\eta=1$ for which a gap
appears between the quark and hadronic branches.
