Most times when the guards come through on the commuter routes I use they will be checking tickets and selling them to those who haven't got them. Sometimes, however, they come through at quite a lick just asking whether anyone 'needs' tickets. They will also often emphasise that they can sell tickets for tomorrow if it is the evening, or renew weekly tickets.
I just wondered how TOCs viewed this? They could be pleased, because the guards are helping commuters and reducing the queues at ticket offices the following morning, but I could also imagine them saying that the idea of giving the guards commission for seeling tickets was in exchange for them checking that everyone had one. The routes concerned would mostly be stations with ticket gates or manned barriers, incidentally.
If someone is stopped at a barrier, it makes it easier to prosecute if they had an opportunity to buy. Also, people will stop the guard and buy to save the hassle of queueing at the excess fares window.
I rarely do it, but on some busy trains you will not get the chance to check every ticket before the next stop. For example, the morning trains from Swansea to Cardiff this week have over 400 passengers on board leaving Bridgend, about 250-300 of them only going to Cardiff, 20 minutes away, which is barriered, so they know they won't get out without a ticket. Also, some South Wales stations will open the barriers and tell everyone in the queue to go through and buy on the train. In this scenario, walking through saying "Who needs one" is the guards only realistic option, then do a full ticket check after Cardiff when it's gone back to normal.
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That said, if a customer asks for it, then you have to sell it.
Though if they are attempting fare evasion by lying about their origin, then you do have the option of reporting them for prosecution instead of selling (a) ticket(s), I believe (depending on the TOC policy on that). If you are saying someone initially lies about their origin and then when sussed, admits it and asks for a split but you want to charge them for the more expensive ticket, you could in theory say "I will sell you the through ticket, with no splits, or alternatively if you do not wish to accept this, I could report you for fare evasion for lying about your origin" however I am unsure if this is good practice or whether the TOC concerned would allow this (it could be seen as blackmail?). It may be on dodgy ground, I don't know.
The only split I usually offer is if someone is going via London, and it's cheaper to split. There are some huge price discrepancies for cross-London tickets as opposed to split, (which I feel are just unfair, IMHO).
I usually don't offer the Didcot split, as it's one that is abused, to be frank (and I'm fed up being abused by people who have to pay as the train doesn't stop there!).
The chances of them being prosecuted for fare evasion in that case is about the same as being struck by lightening, but I will threaten BTP if they do fail the attitude test. Usually, it's a case of sell them the through ticket, and "advise" them that next time do it properly.
