  In a Church synthesis game, two players, Adam and Eve, alternately pick some
element in a finite alphabet, for an infinite number of rounds. The game is won
by Eve if the omega-word formed by this infinite interaction belongs to a given
language S, called the specification. It is well-known that for omega-regular
specifications, it is decidable whether Eve has a strategy to enforce the
specification no matter what Adam does. We study the extension of Church
synthesis games to the linearly ordered data domains (Q, <) and (N, <). In this
setting, the infinite interaction between Adam and Eve results in an omega-data
word, i.e., an infinite sequence of elements in the domain.
  We study this problem when specifications are given as register automata.
Those automata consist in finite automata equipped with a finite set of
registers in which they can store data values, that they can then compare with
incoming data values with respect to the linear order. Church games over (N, <)
are however undecidable, even for deterministic register automata. Thus, we
introduce one-sided Church games, where Eve instead operates over a finite
alphabet, while Adam still manipulates data. We show that they are determined,
and that deciding the existence of a winning strategy is in ExpTime, both for Q
and N. This follows from a study of constraint sequences, which abstract the
behaviour of register automata, and allow us to reduce Church games to
omega-regular games. We present an application of one-sided Church games to a
transducer synthesis problem. In this application, a transducer models a
reactive system (Eve) which outputs data stored in its registers, depending on
its interaction with an environment (Adam) which inputs data to the system.
