Is there a generally accepted term for "Christians in Communion with the Bishop of Rome"? Roman Catholic is a title given to the Latin Rite of the Catholic Church.

By this definition, therefore, the term Roman Catholic explicitely excludes the Eastern Rites of the Church, even though these are fully part of the same religion, in full communion with the Bishop of Rome, and are distinguished only by a different structure to their prayers and services. They have the same beliefs. However, the Anglican Ordinarates, which also have a different liturgical calendar and worship structure, nonetheless count as part of the Latin Rite (and therefore, presumably, as Roman Catholic):

While the personal ordinariates preserve a certain corporate identity of Anglicans received into the Catholic Church, they are canonically within the Latin Church and share the same theological emphasis and in this way differ from the Eastern Catholic churches, which are autonomous particular churches.

Hmm. So what is a generic term for this religion? We can’t call it simply Catholicism, because that has a broader meaning, including Christians who definitely do not accept the authority of the Pope:

I am an Anglican. Specifically, I am an Anglo-Catholic, which means that I believe the Anglican Church is part of the One, Holy, Catholic and Apostolic Church, that its threefold ministry is the threefold ministry of the Catholic Church, and that I am just as Catholic as you are. If you say “Catholics believe the Pope is infallible when speaking ex cathedra”, I’m going to be in the corner with my hand in the air jumping up and down shouting “excuse me, I don’t”.

We seem to have come to an impasse.
Wikipedia resolves the impasse by saying that Roman Catholic is not in fact a synonym of Latin Rite Catholic.

Catholic Church most often refers to:

The Roman Catholic Church, i.e. the Latin Rite (Western) and the 22 Eastern Catholic Churches in communion with the Bishop of Rome (the Pope)
Those churches which, collectively, claim apostolic origins and historical traditions of Catholicism, but not necessarily in the Roman tradition, including the Orthodox and Anglican churches
Most broadly, the Christian Church in general (in English the uncapitalized form of “catholic” is used in this sense; this is the sense that some attach to the phrase in the Apostles’ Creed)


By this definition, the Roman Catholic Church includes both the Eastern and Western branches of the church in communion with the Pope. This contradicts the opening quote. It also gives us an answer to the question: Christians in communion with the Bishop of Rome are Roman Catholics. This makes sense, but is it how the language is actually used? Certainly Peter Turner seems to disagree: he uses the term differently. Perhaps a well-referenced dictionary would be the best source for this. I certainly wouldn’t take the Wikipedia definition as definite fact, not least because it’s hotly disputed on the article’s talk page.
For what it’s worth, Collins gives:

Roman Catholic Church: the Christian Church over which the pope presides, with administrative headquarters in the Vatican Also called: Catholic Church, Church of Rome
Catholic Church:

short for Roman Catholic Church
any of several Churches claiming to have maintained continuity with the ancient and undivided Church


This seems unhelpful.
Perhaps a bit of history concerning the label "Roman Catholic" will serve us well by shedding light upon (if not answering) this very good question.

The term "Roman Catholic" appears to be directly connected to the Reformation in England, and was originally an imposed derogatory term of condescension. 

The Catholic Encyclopedia references the Oxford English Dictionary:


  In the "Oxford English Dictionary", the highest existing authority
  upon questions of English philology, the following explanation is
  given under the heading "Roman Catholic".
  
  The use of this composite term in place of the simple Roman, Romanist,
  or Romish; which had acquired an invidious sense, appears to have
  arisen in the early years of the seventeenth century. For conciliatory
  reasons it was employed in the negotiations connected with the Spanish
  Match (1618-1624) and appears in formal documents relating to this
  printed by Rushworth (I, 85-89). After that date it was generally
  adopted as a non-controversial term and has long been the recognized
  legal and official designation, though in ordinary use Catholic alone
  is very frequently employed. (New Oxford Dict., VIII, 766)


One of the earliest appearances of the term "Roman Catholic" is in a book written by Robert Crowley called "A Deliberat Answere", printed in 1588. Though preferring to use the terms "Romish Catholike" or "Popish Catholike", he also writes of those "who wander with the Romane Catholiques in the uncertayne hypathes of Popish devises" (p. 86). A study of these and other early examples in their context shows plainly enough that the qualification "Romish Catholic" or "Roman Catholic" was introduced by Protestant divines who highly resented the Roman claim to any monopoly of the term Catholic. In Germany, Luther had omitted the word Catholic from the Creed, but this was not the case in England.

It is noteworthy that by the 17th century the label of "Roman Catholic," although a term of condescension, began to be subordinately accepted by both the Catholic clergy, as well as the laity, most prevalantly in heavily persecuted Ireland.


  ...[T]he use of the term Roman Catholic continued to be a mark
  of condescension, and language of much more uncomplimentary character
  was usually preferred. It was perhaps to encourage a friendlier
  attitude in the authorities that Catholics themselves henceforth began
  to adopt the qualified term in all official relations with the
  government. Thus the "Humble Remonstrance, Acknowledgment,
  Protestation and Petition of the Roman Catholic Clergy of Ireland" in
  1661, began "We, your Majesty's faithful subjects the Roman Catholick
  clergy of Ireland".


Up until the turn of the 20th century, Catholic clergy were "strongly advised" to maintain the official title of "Roman Catholic" for the sake of separation from the Church of England.


  In 1897 at the Diamond Jubilee of the accession of Queen Victoria, and
  again in 1901 when Edward VII succeeded to the throne, the Catholic
  episcopate desired to present addresses, but on each occasion it was
  intimated to the cardinal that the only permissible style would be
  "the Roman Catholic Archbishop and Bishops in England". Even the form
  "the Cardinal Archbishop and Bishops of the Catholic and Roman Church
  in England" was not approved. On the first occasion no address was
  presented, but in 1901 the requirements of the Home Secretary as to
  the use of the name "Roman Catholics" were complied with, though the
  cardinal reserved to himself the right of explaining subsequently on
  some public occasion the sense in which he used the words (see
  Snead-Cox, "Life of Cardinal Vaughan", II, 231-41). Accordingly, at
  the Newcastle Conference of the Catholic Truth Society (Aug., 1901)
  the cardinal explained clearly to his audience that "the term Roman
  Catholic has two meanings; a meaning that we repudiate and a meaning
  that we accept." The repudiated sense was that dear to many
  Protestants, according to which the term Catholic was a genus which
  resolved itself into the species Roman Catholic, Anglo-Catholic, Greek
  Catholic, etc. But, as the cardinal insisted, "with us the prefix
  Roman is not restrictive to a species, or a section, but simply
  declaratory of Catholic." The prefix in this sense draws attention to
  the unity of the Church, and "insists that the central point of
  Catholicity is Roman, the Roman See of St. Peter."


All things considered, I believe its safe to say that all English speaking countries refer to Christians in union with the Pope (both Eastern and Western) as "Roman Catholics" mainly because of the trickled down term that has been imposed on papal subordinates by the British Crown. 

Of course there are many other reasons...

Long story short - the Reformation complicated things terminologically for English speaking Christians, as with many other aspects within Christianity. 


  It is noteworthy that the representative Anglican divine, Bishop
  Andrewes, in his "Tortura Torti" (1609) ridicules the phrase Ecclesia
  Catholica Romana as a contradiction in terms. "What," he asks, "is the
  object of adding 'Roman'? The only purpose that such an adjunct can
  serve is to distinguish your Catholic Church from another Catholic
  Church which is not Roman"


I would like to reiterate that, although this may not answer your question satisfactorily, it may at least shed some light...



Personal experience/opinion...

I wish the label of Roman Catholic would be done away with altogether. I remember the first time I encountered this term. I was going through confirmation (around the mid '90s) and had only been taught by friends, family, and fellow parishioners that we were all simply "Catholics." I attended a youth retreat with thousands of other teenagers. I noticed someone wearing a shirt that said "Proud to be a Roman Catholic." I asked my youth leader, "What is a Roman Catholic?" He responded, "Well I guess it is just an emphasis on the fact that the Pope is our Pope and he happens to be the Bishops of Rome. Maybe that shirt is to signify to Protestants that he is marked by the Anti-Christ." (He was joking of course)

The label is self contradictory. If what we say is true...that the Church is all inclusive...then we shouldn't confuse people by clinging to an archaic label that was forced upon us in the first place. Perhaps "Petrine Catholics" or something of that nature would be more fitting. This question should not exist.

Maybe we should all write a letter to his Holiness suggesting a better label for the world to recognize.