How to register Russian tourist visa while on the Trans-Mongolian railway?

The legal technicalities are as follows:

• What is registered is an alien’s stay (not visa). While an alien is en route, no legal obligation of being registered exists.
• It is hosts’ duty to register their alien guests. The latest edition of the law on registration even explicitly states that under no circumstances aliens may be held responsible for violations (except when the alien and the host are the same person).
• Hotels and alike must register their guests within 24 hours after arrival (not counting public holidays).
• Private hosts must register their guests within 7 business days after arrival (not counting the day of arrival itself). Increased deadlines apply for nationals of some CIS countries.
• Hosts register their alien guests by sending a notification form by mail at a post office. The post office certifies the ‘detachable part’ of the form and hands it to the host as a proof. How large the fee is nowadays I don’t know but suppose it must be somewhere around 300 RUB.
• Upon entry onto the territory of the Union State of Russia and Belarus, an alien fills a so called ‘migration card’. The departure half of the card must be turned in at exit. However, the old practice when registration records were stamped on the card was abolished a few years ago (in Russia, but not in Belarus). Since then, there was also the stipulation in law that upon departure from a place of stay, aliens had to leave their ‘detachable parts’ to their hosts so that they turned them in to the authorities. This has also been canceled, though there is no clarity as to if aliens now have to keep them until exit.

Practicalities are a bit more complicated due to the variety of interpretations.

• Generally, Russia suffers from a serious degree of lawlessness. Many policemen have a very vague notion about what the law is (which, nevertheless, never prevents them from acting), and judges may easily hold you responsible for things which the law explicitly states you cannot be held responsible for. However, registration laws are mostly turned against those who live or do business in Russia (and are therefore ‘easy prey’) or, as we say, ‘crossed the road’ of someone influential. I haven’t heard about ordinary tourists being harassed for quite a long time.
• Despite all the wordings of the law, it is quite widely presumed that an alien must be registered within 7 days since entering the country or leaving the previous place of registration.
• Large ‘well-established’ hotels register their guests easily. Many of them don’t even require the official fee. However, there have been reports of some weird cases, such as when a hotel in Nizhniy Novgorod refused check-in to tourists who had no proofs of registration after 7 days since entering the country.
• Small hotels and especially hostels often operate ‘in grey area’ and more often require fees above the official ones. Those of them who cater for international clientele, will register you willingly; in any case, they need a day to do that.
• Passport control officers, at least in large airports, never ask anyone to produce proofs of registration.

Thus, to play safe without being paranoid:

• Most probably, you will have no problems at all even if you never are registered. Don’t give too much effort and thought to the issue.
• For greater confidence, if you stay somewhere longer than one night, ask about registration. Registering just once before you get to Moscow will be sufficient.
• Insist on registration at all places where you stay a substantial span of time.
• If you look like tourists, it is highly unlikely that you are approached by the police. However, if you are, be polite but don’t give in if you smell something fishy. Don’t be proactive, only show as much as requested. Have your nearest consulate number in your telephone, and when in doubt about anything, call them for advice. Russian words you may need for communication in these circumstances are mostly adopted from Latin or French: ‘palitsiya’ (police), ‘dakumenty’ (documents), ‘paspart’ (passport), ‘viza’ (visa), ‘migratsionnaya karta’ (migration card), ‘registratsiya’ (registration), ‘bilet’ (ticket), ‘telefon’ (phone), ‘konsul’ (consul).
• Keep your tickets. Most probably, you won’t need them, but ‘just in case’ presenting them may be the easy way to quickly refute any attempts of accusations.