  Classically, any structure for a signature {\Sigma} may be completed to a
model of a desired regular theory T by means of the chase construction or small
object argument. Moreover, this exhibits Mod(T) as weakly reflective in
Str({\Sigma}).
  We investigate this in the constructive setting. The basic construction is
unproblematic, however, it is no longer a weak reflection. Indeed, we show that
various reflection principles for models of regular theories are equivalent to
choice principles in the ambient set theory. However, the embedding of a
structure into its chase-completion still satisfies a conservativity property,
which suffices for applications such as the completeness of regular logic with
respect to Tarski (i.e. set) models.
  Unlike most constructive developments of predicate logic, we do not assume
that equality between symbols in the signature is decidable. While in this
setting, we also give a version of one classical lemma which is trivial over
discrete signatures but more interesting here: the abstraction of constants in
a proof to variables.
