  Charge stripe order has recently been established as an important ingredient
of the physics of cuprate high-T$_c$ superconductors. However, due to the
complex interplay between competing phases and the influence of disorder, it is
unclear how it emerges from the parent metallic state. Here we report on the
discovery of an unconventional electronic ordered phase between charge-stripe
order and (pseudogapped) metal in the cuprate La$_{2-x-y}$Eu$_{y}$Sr$_x$CuO$_4$
with $y = 0.2$ (LESCO). The peculiar properties of the intermediate phase are
revealed through three complementary experiments: nuclear quadrupole resonance,
nonlinear conductivity, and specific heat. We demonstrate that the order
appears through a sharp phase transition, and exists in a dome-shaped region of
the phase diagram, similar to charge stripes. A comparison to recent
theoretical work shows that the order is a state without broken translational
symmetry -- a charge nematic. We thus resolve the complex process of charge
stripe development in cuprates, show that the nematic phase is unrelated to
high-temperature pseudogap physics, and establish a link with other strongly
correlated electronic materials where nematic order is prominent.
