The topologists are very short of ZFC examples of
non-metrizable perfectly normal compact spaces. The most daring
hypotheses
about this class persist for dozens of years without noticeable
progress in
their solution. In fact all perfectly normal compact spaces known in
ZFC
are some derivatives of the double arrow space S (such as continuous
images of closed subsets of 
, where I is the unit
segment [0,1] with its usual topology). That's why D.H.Fremlin asked
if it was consistent that any perfectly normal compact space has a
two-to-one continuous map onto a metrizable one. This question was cited by
G.Gruenhage in [5].
It seems to be a folklore that no Souslin continuum with all its intervals non-separable admits a continuous map onto a metrizable space with the inverse images of all points metrizable. This clearly implies that the negative answer to D. H. Fremlin's question is consistent with ZFC.
If a space X can be mapped continuously and with metrizable fibers
onto a
metrizable space, we say that X is metrizably fibered. We take a
look
at the class 
of compact metrizably fibered spaces. It is
proved that is strictly smaller than the class 
of
first
countable compact spaces. We show that is invariant under
open
maps, but not under the continuous ones. We establish, however, that
if X
is a perfectly normal compact space from , then any continuous
image
of X belongs to too. We also introduce the class of weakly
metrizably fibered spaces and prove that each Eberlein compact space
of weight less than or equal to continuum belongs to it.