Short Answer: Generally No.
I am not aware of any English speaking countries that have such a court, which would sometimes be translated as a "housing court" in comparative law descriptions of court systems that have them.
Incidentally, the term "co-tenants" would ordinary be used for co-owners of a single piece of real estate, or co-renters of a single piece of leased real estate, but not to refer to fellow members of a condominium association.
Normally, in legal systems based on the common law of England, cases like this are handled in the same civil trial courts as breach of contract cases, property disputes and personal injury cases. In my own state of Colorado, covenant violations and small debt collections can be brought in a court of limited jurisdiction called a county court that also handles evictions and money claims up to $15,000 and temporary restraining orders, while other disputes are brought in a court of general jurisdiction called a district court.
I have seen charts of organization for court systems in England, Scotland and New Zealand in the past, as well as all U.S. jurisdictions, and don't recall any courts specifically devoted to condominium disputes in any of them.
Even in civil law countries, specialized courts with jurisdiction over condominium disputes are quite uncommon as I have seen organizational charts for many of those. I seem to recall one or two in the Benelux countries, however.
Alternative Dispute Resolution Programs For Condominium Owners
Some condominium associations, however, do have an arbitration agreement included in them that requires that such disputes be referred to a private arbitrator, although binding arbitration provisions aren't very common in condominiums in most parts of the U.S.
Florida recently adopted mandatory non-binding arbitration of condominium disputes which is a pre-requisite to taking a condominium case to court.
Connecticut has or is contemplating a pilot condominium dispute mediation program as part of the services provided by its probate court, but rejected adding jurisdiction over these cases to its general purpose small claims or limited jurisdiction court as Colorado does. Mediation is a form of non-binding settlement discussion which is facilitated by a third party mediator.
I also review below several housing courts in the U.S., none of which appear to clearly have condominium dispute jurisdiction.
Stand Alone Housing and Land Courts
Massachusetts has a stand alone court whose jurisdiction over condominium matters is at best not well advertised and at worst absent or unclear. Massachusetts also has a Land Court, but that also doesn't seem to have jurisdiction (it handles foreclosures, partition actions, and zoning cases). It is the only state of which I am aware that has a stand alone housing court as opposed to a specialized division of another more general purpose court.
The City of Providence, Rhode Island has a housing court on the Massachusetts housing court model with similarly vague jurisdiction, established as part of its authority to create municipal courts, rather than as part of a state court system as in Massachusetts, but not as a mere division of a municipal court either. Again, it probably lacks condominium dispute jurisdiction.
Housing Courts As Divisions Of General Purpose Courts
New York City has a division of the limited jurisdiction city court for New York City handles such cases (up to $25,000 in controversy) but I don't know if it has jurisdiction over condominium disputes or just landlord-tenant and housing code violations. Cleveland similarly has a division of its municipal court called housing court with similar jurisdiction that also doesn't appear to have jurisdiction over condominium disputes, just housing code violations and landlord-tenant disputes. There is a housing court division in Superior Courts in San Francisco, California (which also seems to lack condominium dispute jurisdiction although it isn't entirely clear).
Housing Courts In Name Only
The City of Baltimore, Maryland appears to use the term "housing court" more figuratively to reference to the docket of cases brought by city officials in the housing department, rather than as a true administratively distinct court.