Yes they are very careful to call it "foreign law" when in reality it is religious law. You will never see a Republican write a bill outlawing religious law in our government or court rooms. Sharia law is religious law. There are many natural born citizens that adhere to Sharia law.
A Tribunal is not a real court. Those that CHOOSE to allow a tribunal to decide an outcome are free to do so. If they decide they want a real court of law to rule on an issue then we already have in place laws (and constitutional amendments) that assure rulings based on actual US laws. These Republicans that are pushing for the bans (like we have in Alabama) are just pandering to their nutty constituents that think they will somehow be forced into a Sharia law tribunal.
The Catholics have a Christian law Tribunal. Of course that is different now isn't it. 
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Points well taken Jankinonya! Some universities have student courts as part of the university's disciplinary system. Students upon entering the university are apprised of this arrangement and thus they voluntarily accept it. Such courts no more displace the civil law of federal, state, or local laws that does Sharia law when it is accepted as a means of resolving civil disputes between disagreeing parties. Of course, NO law other than federal, state or local, can prescribe penalties such as imprisonment or death to offenders or otherwise impose criminal penalties that would in themselves be crimes under the prevailing statutes of governments. The Texans who are all bent out of shape about Sharia law are just another subset of Islamaphobic wingnuts.