The rules of MWBCs
I understand this is NOT to code since there is no tie bar between the two breakers (confirm this WOULD be to code if this was the case).
Correct. This is called a "Multi-Wire Branch Circuit" or MWBC. Yes, circuit singular. A MWBC is one circuit for safety purposes. (though for for the provisioning rules, you can treat them as separate, e.g. 1 MWBC an power 1 kitchen receptacle "circuit" and 1 bathroom "circuit").
You must follow several rules with those:
- As Doxylover discusses, the two hots must be on opposite poles, which would mean 240V between the two hots. This is so neutral carries only difference current, and isn't overloaded. 0V between hots is BAD, which is why a duplex/tandem breaker cannot be used as the 2 breakers (there would be 0V between the wires since they are on the same pole/phase).
- As you said, the handles must be tied. This must be an approved handle tie, meaning made by the manufacturer and UL-Listed for tying those breakers. Not allowed to use a nail - after all, a nail could be used to tie the 2 handles of a tandem/duplex!
- In each junction box, neutral must be pigtailed. You are not allowed to use devices to make a neutral splice (e.g. the 2 screws on a typical receptacle, 1 for supply and 1 for onward wire). That is because the neutral cannot be broken if a device is removed (or there'd be very bad consequences downline). Ground already follows this rule, you may notice.
Handle ties: yes. Tandems are a complication.
Should the panel be rewired where this condition exists, and there are several in here, land next to each other at the panel then simply use tandem connected breakers with tie bars?
Well not TANDEMS, for Pete's sake!
But yes, this needs to be done. Leaving a note is a terrible idea, most people have no idea what a MWBC is and will "don't understand, don't care".
I would also argue that moving breakers around is hardly "rewiring", it should be super easy. Worst thing you might need is a few wire nuts and extra wire for pigtails, in case the provided wire is not long enough. (some builders obsess with clipping all wires as short as possible, bad idea).
However all your breakers are "double-stuffs" except a few 2-pole breakers. So this is a problem because you need to find a way to do it with tandems after all. Doxylover covers one way to do it: with Quadplex breakers, which are paired tandems that correctly access both poles and give an "inner" and an "outer" 240V breaker. Those will work nicely.
The other way is manufacturer-specific. Some have a "handle tie" that works between TWO tandem breakers (so 4 throws). **This "ties" the inner handle on each breaker, sort of like a quadplex with only the inner handles tied.
You can stack those indefinitely- so you can go like this
2A 2B--3A 3B--4A 4B--5A 5B--6A 6B
Giving two single breakers (2A and 6B) and four handle-tied pairs. There's a practical limit to how many you can hold together when fitting it into the panel, LOL!
However you must use THE correct breaker type for your panel. Breakers are NOT interchangeable across brands. I think in the 60s they were copying each other's designs, but patent infringement suits forced everyone to make their bus stabs a different shape. As a result, alien breakers won't engage the bus properly, and will arc and destroy the bus stab, if not the panel, if not the house.
Get us any information off the stickers off your panel - there may be important stickers on the backside of the deadfront cover. (the one that does not fit properly). From that we can identify the proper modern breakers approved for your panel. They don't cost a lot, $5 per pole before the weird shortages started happening.
Some other notes.
The panel cover seems to be fitting extremely poorly. I imagine you've had a few outlet and switch boxes open in your career, and you've noticed the box is often sunk into the wall, and the system is designed to accommodate that. And you see an sunken service panel and you think "that must be fine too". Absolutely not. Service panels cannot be sunk. The lid won't fit properly and the breakers will fall out. The cover's proper positioning is the ONLY thing that keeps breakers in place. The edge of the breaker needs to catch the lid.
So do whatever - mill out the interfering stucco, or shim the panel out so it's flush. That lid needs to go on properly.