It's not entirely clear if the number Cloudsim displays as the caller ID is the temporary number you've been assigned, or your actual real phone number. If it's the latter, that's a big problem, NEVER give an escort access to an actual phone number registered in your name! But it would be strange for Cloudsim to do that. Can someone check?
Even if it's the temporary number (which seems more likely), you have to weigh the risk of your wife/girlfriend finding a separate punting phone (or sim), versus the risk of either your wife/girlfriend finding the app on your phone, or a malicious escort tracing you through it. Presumably if she makes a bogus allegation against you to get the police involved, they could ask Cloudsim for your real number anyhow.
Anonymity for punters is needed for two reasons: to hide your activities from your partner, and to protect you from "rogue" escorts (blackmailers or crazies: something I unfortunately have experience of, so the danger IS real). An unregistered punting phone could be found by your partner, but in the rogue escort situation, even the police would have difficulty tracing it to you (and they probably wouldn't bother unless the accusation against you was very convincing: as they're known to monitor UKpunting, I doubt that they would act on an allegation from someone known to be dodgy, but you never know).
