In fact there are only two verdicts: guilty and not guilty. What you describe is a jury that cannot reach a verdict. In such cases, the judge is almost always able to talk them into further deliberations. However when they stand rock solid and say "no more"the judge will accept the that the jury is deadlocked, send them home and declare a mistrial.
After a mistrial is declared by the court, the prosecution will decide whether to retry the case. If the swing was in favor of acquital, the prosecution may hesitate to retry. If it was 9-3 to convict, they almost certainly will retry the matter.
When a jury is unable to come up with a unanimous verdict, the Court will usually make them go back and deliberate once or twice before finally accepting that they are "deadlocked". Once the Court has determined that a jury cannot (and will not, even with further deliberations) reach a unanimous verdict, a mistrial will be declared and the jury will be sent home. The Court will then set a future date. At this future date, the prosecutor announces whether or not they will seek to retry the case. Prosecutors look at many factors when making this decision, but the primary factor they look at is how the jury was split. If the jury is 10-2 for acquittal, it is unlikely the prosecutor will seek to retry the case. If it is 10-2 for conviction, the DA will most certainly retry it. If the prosecutor seeks to retry the case, the defense will usually "invite" the Court to dismiss the case on the Court's own motion. Depending on how the judge rules, the case will proceed to another trial.
Judges like verdicts because it means one less case on their calendar, so they will usually encourage jurors to continue deliberations and try to reach a verdict. Usually there is no announcement of the verdit in the courtroom. The judge will usually ask what the split is to see how close the jury is to a verdict, but they will not ask which way the majority is leaning. The closer the split, the more likely the judge will ask the jury to keep deliberating, at least for a while, until the judge reaches the conclusion that obtaining a unanimous verdict is not going to happen. A unanimous verdict is deisred by the court but is not required. Anything less than unanimity leaves the door open for a new trial without violating the rules against double jeopardy. Technically the case can be tried over and over again although the judge can dismiss the case in the interest of justice if the judge believes a new trial would be a waste of time and judicial resources -- trials are very expensive.
A verdict is by definition, unanimous. Thus, 9-3 (in a criminal matter) means the jury is deadlocked at that point. The judge sure can make the jury go back and deliberate further unless and until he or she is satisfied that there is no reasonable likelihood that the jury can all agree. If the jury stays deadlocked, whether 9-3, 8-4, 6-6, 11-1, and there is no way the jurors will come to a verdict, then we have a hung jury. A hung jury is typically a win for the defense even though the case starts all over for many reasons I can explain if you are interested further. Note, that a defense lawyer should pop up and move for a mistrial as soon as it becomes reasonably clear that the jury is deadlocked EVEN if the judge denies and keeps the jury in. Note, that judges do not want a hung jury mistrial because they want the case to resolve itself and be over in one way or another [acquittal, settlement, or conviction] ... that is why you may be seeing or hearing the judge pushing the jury to deliberate further after the jury has already written the judge that they are deadlocked.
If the jurors are very strong in delivering to the judge that they have 9 to 3 decision, the judge will declare a mistrial because the jurors can only come with either "guilty" or "not guilty" verdict; he or she will try to talk them into deliberating more and come with unanimous verdict (most judges will send the jurors back to the jurors room for further deliberations) and order them back to the jurors room in hope that they eventually come with "guilty" or " not guilty" verdict. If after a few tries, the jurors are still 9 to 3, the judge will not have any other options, but to declare mistrial. But it is not the end of the case yet - in misdemeanor trial, it would be because in misdemeanor jury trials, if mistrial is declared the DA cannot retry the case. The DA will decide further whether to retry the case or not. It all depends on whether 9 is in favor of 'not guilty' or 'guilty' verdict. But the DA will also have to look at other factors - the strength of the DA's case, credibility of the witness(es), the seriousness of the offense(s), the time it takes to go to jury trial, the case load the DA;s office had and the cost to the DA and the county to have the case go to jury trial again (and most counties have serious budget cuts now), involvement of experts; so it all depends on many factors.