  By combining theory and experiments, we demonstrate that dipolar quantum
gases of both $^{166}$Er and $^{164}$Dy support a state with supersolid
properties, where a spontaneous density modulation and a global phase coherence
coexist. This paradoxical state occurs in a well defined parameter range,
separating the phases of a regular Bose-Einstein condensate and of an
insulating droplet array, and is rooted in the roton mode softening, on the one
side, and in the stabilization driven by quantum fluctuations, on the other
side. Here, we identify the parameter regime for each of the three phases. In
the experiment, we rely on a detailed analysis of the interference patterns
resulting from the free expansion of the gas, quantifying both its density
modulation and its global phase coherence. Reaching the phases via a slow
interaction tuning, starting from a stable condensate, we observe that
$^{166}$Er and $^{164}$Dy exhibit a striking difference in the lifetime of the
supersolid properties, due to the different atom loss rates in the two systems.
Indeed, while in $^{166}$Er the supersolid behavior only survives a few tens of
milliseconds, we observe coherent density modulations for more than $150\,$ms
in $^{164}$Dy. Building on this long lifetime, we demonstrate an alternative
path to reach the supersolid regime, relying solely on evaporative cooling
starting from a thermal gas.
