The National Electrical Code defines a Multi-wire branch circuit as follows.

Branch Circuit, Multiwire. A branch circuit that consists of two or more ungrounded conductors that have a voltage between them, and a grounded conductor that has equal voltage between it and each ungrounded conductor of the circuit and that is connected to the neutral or grounded conductor of the system

Background

In the US, residential dwellings are often supplied using a 120/240V single split-phase system. This system consists of two ungrounded (hot) conductors, and a grounded (neutral) conductor (which is a center tap on the transformer).

Multi-wire branch circuits

A Multi-wire branch circuit takes advantage of this, by extending both ungrounded conductors and sharing a single grounded (neutral) conductor. So instead of having two circuits requiring four conductors, you can have two circuits with only three conductors.

How to recognize them

Multi-wire branch circuits will typically be supplied by two adjacent breakers in a panel, and will often be fed using 3-wire plus ground cable.

If the circuit was installed properly, the breakers handles should be tied together or a double pole breaker will be used.

What to worry about

• If you're working on the circuit, you'll want to make sure both breakers are off.
• You should never disconnect the grounded (neutral) conductor, while the breakers are on.
• You'll have to take care installing GFCI devices, as it can be a bit tricky.
• You'll have to make sure the ungrounded conductors come from separate legs of the service.

AFCIs are also a bit like GFCIs in that they care a lot about crossed neutrals. A LOT. You will never get any AFCI to hold while a crossed neutral persists, nor should you! Remember, neutrals do not have breakers, the only thing that keeps them from overloading is the fact that they are fed from only one hot. If they are fed by 3 hots, one could return 45A and of course start a fire in the walls.

You will have to identify every outlet (including hardwired ones) on each circuit. Pull the covers off and the devices out. Document how they are wired. It is likely you will uncover the problem at that point.

If not, disconnect them all and light up each circuit, one at a time, to determine which is the "first" outlet. Hook up that outlet correctly (as opposed to how it is now) and test. Continue one at a time, thoroughly testing each time.

When you run out of things to hook up, and you still don't have problems, then you have licked it. Otherwise, the problem will return at a particular step. That is your problem area.

You seem sanguine about some crossed circuits in a 3-way light circuit. That may be one of your problems, and there may not be a legit way to regain original functionality without going to smart switches.