Can a slow and strong phase transition in neutron stars relieve major compact-star observation tensions?

Recent anomalous compact-star observations challenge the conventional neutron-star interpretation in complementary ways: HESS J1731--347 and XTE J1814--338 favor unusually small radii at low mass, while the secondary component of GW190814 appears too massive for an ordinary neutron star under GW170817-based maximum-mass inferences. We examine whether neutron stars with a strong first-order hadron-quark phase transition can address these tensions via the extended stable hybrid branches that arise when the phase conversion is slow compared to radial oscillations, while remaining consistent with GW170817 and NICER constraints. Using a piecewise-polytropic (PP) benchmark, supplemented by an independent speed-of-sound (CS) parametrization comparison, we find two viable scenarios after a general parameter scan over both the hadronic and hybrid branches. Scenario 1 realizes an all-at-once solution: the same equation of state has a pure hadronic branch compatible with both the GW190814-scale mass and HESS J1731--347, while its slow stable hybrid branch reaches XTE J1814--338. Scenario 2 retains the GW190814-scale hadronic branch but reaches XTE J1814--338 and HESS J1731--347 on slow stable hybrid branches in very different transition-strength regimes.