Resolving Conflicting Injunctions
This happens from time to time, frequently in cases involving child custody with dueling cases in different states, at least until recent legislative changes (the Uniform Child Custody Jurisdiction Act and the Parental Kidnapping Prevention Act), which set forth clear rules calculated to leave only one court with jurisdiction to enter orders in almost all cases. The very first case I worked on as a student in a law school clinical program involved this kind of jurisdictional standoff.
There is no fool proof way to prevent this, although parties will often forcefully ask the court not to have contradictory injunctions imposed upon them, and courts will often take these arguments seriously. And, the parties will usually parse the language of each order to determine how both can be possibly be complied with at once.
Also, impossibility is a defense to being held in contempt of court for violating an injunction, and this tends to favor honoring the first entered of contradictory injunctions since the first is not impossible to comply with but complying with subsequent orders may be impossible.
When push comes to shove and the appeals in each case don't change the result, usually the case keeps going up the appellate court ladder on an emergency basis in each contradictory case until it the deadlock is resolved, or until a court with jurisdiction over all courts involved intervenes.
But, as you note, there is no definitive way to prevent the highest court in the land from having a deadlock (which can happen even when there are an odd number of judges if a judge recuses).
In many state supreme courts, and in appellate courts outside the U.S., deadlock in a tie breaker court is often avoiding by having a process to appoint lower court judges a substitute judges in a higher court when there is a potential for deadlock in that court. But, the U.S. Supreme Court has no such process.
The Ongoing Policy Debate
There is on ongoing debate in federal public law practice (i.e. in cases in which the government is a party) over whether national injunctions of the type that have been issued in the DACA case and the Muslim ban cases in the current administration are proper, or not on jurisdictional grounds (but, the propriety of the injunction is generally not a defense to being held in contempt of court for violating it). These injunctions bind the United States government, which is a single entity over which the court has jurisdiction, but the dispute is over whether these injunctions should be allowed to benefit non-parties who are similarly situated to those before the court (the prevailing rule at this time), or whether the parties seeking an injunction against the United States government only have standing to obtain relief for themselves personally, in the absence of a class action under Federal Rule of Civil Procedure 23, which has procedures calculated to protect the interests of members of a class who are not personally before the court.
The "conservative" position is that "national" injunctions against the United States should not be allowed outside of class actions, which would reduce the potential risk of conflicting orders, but would allow the United States government to proceed with conduct that had already been judicially determined elsewhere in a case against the same defendant to be unacceptable and illegal conduct. The status quo of allowing "national" injunctions is also important to allow a court to have the power to grant relief to individuals whose current location is not known perhaps, for example, because the United States government has concealed this information.
For example, a national injunction allows a district court to grant relief to a parent whose child has been removed from the parent improperly to an unknown location in the juvenile immigration detention system for which the U.S. government lost track of 1,400+ juveniles due to poor record keeping.