Can heterogeneous prices be consistent with competitive equilibrium? I do not want to put many constraints on the structure of the problem, except by defining a competitive equilibrium as such where firms are price takers in the factor and good(s) market, and, perhaps (?), that firms make no profits.

For example, is it possible to have something like "competitivistic" competition, where consumers have love for variety and firms produce differentiated goods, yet, instead of monopolistic competition, firms do not charge a markup, because there is free entry, so that firms end up producing at zero profit?

Or for example, a model of consumers geographically immobile, and where differentiated transportation costs across regions mean firms need to charge a markup to pay for that cost, and where, again, free entry deter firms from exploiting consumers?
As a starting point, you might take a look at Stigler's model of price dispersion. Quoting from a resource I found online,


  George Stigler’s 1961 “The Economics of Information” begins with: 
  
  "One
  should hardly have to tell academicians that information is a valuable
  resource: knowledge is power. And yet it occupies a slum dwelling in
  the town of economics." 
  
  Stigler documents the existence of price
  dispersion empirically, and proceeds to sketch various theories that
  might be applicable to price dispersion. Stigler is careful to
  distinguish price dispersion, which results from imperfect information
  on the part of consumers, from price discrimination, which involves
  distinct prices for different types of consumer preferences
  
  There are various reasons that price dispersion might arise. Clearly
  some consumers must be willing to pay higher prices than others, that
  is, distinct consumers have distinct reservation prices. (Note that a
  reserve price – the minimum bid in an auction – is a very different
  concept.) Differences in reservation prices can arise because of
  differences in knowledge – some consumers have access to price lists,
  others don’t – or search costs. If some consumers have lower search
  costs, then these consumers will search for lower prices.


You might also check out the Wikipedia articles on Price Dispersion and Search Theory.