  It is usually thought that a single equation of state (EoS) model "correctly"
represents cores of all compact stars. Here we emphasize that two families of
compact stars, viz., neutron stars and strange stars, can coexist in nature,
and that neutron stars can get converted to strange stars through the
nucleation process of quark matter in the stellar center. From our fully
general relativistic numerical computations of the structures of fast-spinning
compact stars, known as millisecond pulsars, we find that such a stellar
conversion causes a simultaneous spin-up and decrease in gravitational mass of
these stars. This is a new type of millisecond pulsar evolution through a new
mechanism, which gives rise to relatively lower mass compact stars with higher
spin rates. This could have implication for the observed mass and spin
distributions of millisecond pulsars. Such a stellar conversion can also rescue
some massive, spin-supported millisecond pulsars from collapsing into black
holes. Besides, we extend the concept of critical mass $M_{\rm cr}$ for the
neutron star sequence (Berezhiani et al. 2003; Bombaci et al. 2004) to the case
of fast-spinning neutron stars, and point out that neutron star EoS models
cannot be ruled out by the stellar mass measurement alone. Finally, we
emphasize the additional complexity for constraining EoS models, for example,
by stellar radius measurements using X-ray observations, if two families of
compact stars coexist.
