  We study synthesis of reactive systems interacting with environments using an
infinite data domain. A popular formalism for specifying and modelling such
systems is register automata and transducers. They extend finite-state automata
by adding registers to store data values and to compare the incoming data
values against stored ones. Synthesis from nondeterministic or universal
register automata is undecidable in general. However, its register-bounded
variant, where additionally a bound on the number of registers in a sought
transducer is given, is known to be decidable for universal register automata
which can compare data for equality, i.e., for data domain (N,=). This paper
extends the decidability border to the domain (N,<) of natural numbers with
linear order. Our solution is generic: we define a sufficient condition on data
domains (regular approximability) for decidability of register-bounded
synthesis. The condition is satisfied by natural data domains like (N,<). It
allows one to use simple language-theoretic arguments and avoid technical
game-theoretic reasoning. Further, by defining a generic notion of reducibility
between data domains, we show the decidability of synthesis in the domain
(N^d,<^d) of tuples of numbers equipped with the component-wise partial order
and in the domain (\Sigma^*,\prec) of finite strings with the prefix relation.
