  Context. Asteroids with semi-major axes very close to that of a host planet
can avoid node crossings when their nodal points are at perihelion and at
aphelion. This layout protects the asteroids from close encounters, and
eventual collisions, with the host planet.
  Aims. Here, we study the short-term dynamical evolution of four recently
discovered near-Earth asteroids (NEAs) --2012 FC71, 2014 EK24, 2014 QD364, and
2014 UR-- that follow very Earth-like orbits.
  Methods. Our analysis is based on results of direct N-body calculations that
use the most updated ephemerides and include perturbations from the eight major
planets, the Moon, the barycentre of the Pluto-Charon system, and the three
largest asteroids.
  Results. These four NEAs exhibit an orbital evolution unlike any other known
near-Earth object (NEO). Beyond horseshoe, tadpole, or quasi-satellite
trajectories, they follow co-orbital passing orbits relative to the Earth
within the Kozai domain. Our calculations show that secular interactions induce
librations of their relative argument of perihelion with respect to our planet
but also to Venus, Mars, and Jupiter. Secular chaos is also present. The size
of this transient population is probably large.
  Conclusions. Although some of these NEAs can remain orbitally stable for many
thousands of years, their secular dynamics are substantially more complicated
than commonly thought and cannot be properly described within the framework of
the three-body problem alone owing to the overlapping of multiple secular
resonances. Objects in this group are amongst the most atypical NEOs regarding
favourable visibility windows because these are separated in time by many
decades or even several centuries.
