  We explore systematically a new class of two-phase equations of state (EoS)
for hybrid stars that is characterized by three main features : (1) stiffening
of the nuclear EoS at supersaturation densities due to quark exchange effects
(Pauli blocking) between hadrons, modelled by an excluded volume correction,
(2) stiffening of the quark matter EoS at high densities due to multiquark
interactions and (3) possibility for a strong first order phase transition with
an early onset and large density jump. The third feature results from a Maxwell
construction for the possible transition from the nuclear to a quark matter
phase and its properties depend on the two parameters used for (1) and (2),
respectively. Varying these two parameters one obtains a class of hybrid EoS
that yields solutions of the Tolman-Oppenheimer-Volkoff (TOV) equations for
sequences of hadronic and hybrid stars in the mass-radius diagram which cover
the full range of patterns according to the Alford-Han-Prakash classification
following which a hybrid star branch can be either absent, connected or
disconnected with the hadronic one. The latter case often includes a tiny
connected branch. The disconnected hybrid star branch, also called "third
family", corresponds to high-mass twin stars characterized by the same
gravitational mass but different radii. We perform a Bayesian analysis and
demonstrate that the observation of such a pair of high-mass twin stars would
have a sufficient discriminating power to favor hybrid EoS with a strong first
order phase transition over alternative EoS.
