How does an employer decide which employee to dismiss where there is a personality clash?

The legal issue that would arise here is whether the dismissal of the selected employee is fair and otherwise lawful (for example, not discriminatory).

In terms of fairness of dismissal, it is likely that the employer would need to rely on some other substantial reason (SOSR) as the potentially fair reason for dismissal.  Personality clashes or irreconcilable differences between colleagues can amount to SOSR. As you will see, a tribunal would need to be satisfied that the conflict was causing substantial disruption to the business, and that alternatives to dismissal had been exhausted.  

The more difficult question is how the employer decides which employee to dismiss. We're afraid that we cannot find any authority on this point, although it is likely to be a highly fact-sensitive question in any event. If the situation is genuinely untenable, there is equal fault on both sides, and there are no alternatives, then a tribunal will have some sympathy with an employer who is considering dismissing one or both employees. The employer would, we think, need to consider very carefully its reasons for choosing one over the other (or for electing to dismiss both) and scrutinise those reasons for any hint of discrimination. If the personality of one of the individuals impacts upon working relationships with other colleagues too then that could help to justify the dismissal of that employee over the other. If the dispute is limited to the two individuals concerned, then other factors such as business need may be regarded as relevant.