In reactive synthesis, the goal is to automatically generate an implementation from a specification of the reactive and non-terminating input/output behaviours of a system. Specifications are usually modelled as logical formulae or automata over infinite sequences of signals ($\omega$-words), while implementations are represented as transducers. In the classical setting, the set of signals is assumed to be finite. In this paper, we consider data $\omega$-words instead, i.e., words over an infinite alphabet. In this context, we study specifications and implementations respectively given as automata and transducers extended with a finite set of registers. We consider different instances, depending on whether the specification is nondeterministic, universal or deterministic, and depending on whether the number of registers of the implementation is given or not. In the unbounded setting, we show undecidability for both universal and nondeterministic specifications, while decidability is recovered in the deterministic case. In the bounded setting, undecidability still holds for nondeterministic specifications, but can be recovered by disallowing tests over input data. The generic technique we use to show the latter result allows us to reprove some known result, namely decidability of bounded synthesis for universal specifications.