I think Oktas8 has the story in a reasonable manner.
each degree of bank angle towards the good engine your Vmc reduces by approximately 3 kts
In the right ballpark.
That is, banking towards inoperative engine, your Vmc might get pretty high, right?
Indeed. Banking the wrong way, typically, sees deltas in the 30-40 kt range ... which is why, if you really must be back someone near Vmc, you need to get some favourable bank into the equation VERY promptly .. lest you see the world from an unusual vantage point.
Reading some VMC theory, I don't get it how VMC changes so much if you bank.
Wings level, there will be a residual slip OEI. This can be increased/reduced/removed/made to go the other side according to what bank angle is applied.
Why is slip important ? -
(a) too much might end up putting you in a departure situation and subsequent spin, ergo, let's keep it under some sort of constraint
(b) slip will improve/reduce directional control (ie help/hinder rudder activity) depending from which side the slip happens to come, ie, slip will produce a turning moment in the direction of the slip angle ("further effect of controls" comes to mind from basic training).
(c) wings level, we start with slip towards the dead engine. This is due to the rudder's causing an unbalanced side force .. which accelerates the aircraft slightly to the side ... resulting in a slip in that direction ie the wind now comes from a direction slightly to the dead side of straight ahead
(d) if you were to put full rudder in .. then you can cause yawing in either direction by slip control. I'm not recommending this but, hopefully, you get the idea, ergo, Vmc is VERY sensitive to slip angle (which we, roughly, relate to bank angle)
(e) different consideration to performing a reasonably balanced turn where the slip is kept well under control
in this case the recovery is to reduce all the power on the operating engine and to hope you haven't already strike the ground
Important points to keep in mind for if you get yourself into that situation.
you can't go back to wings level using opposite aileron
That will depend on Type and circumstances on the day. Better to keep a healthy margin above Vmc and leave that regime of flight to the test pilot folks.
Generally, Vmc is determined statically (a bit like the endorsement Vmc demo) and then checked for dynamic effects. Result is that a competent pilot ought to be able to keep the aircraft blue side up and recover to a sensible flight condition
if below Vmc you still have control to oppose the roll and to level the wings back or assume the 5 degrees bank into the operative and so reaching again the attitude where you can maintain straight flight.
Below Vmc you will have great difficulty with directional control at high thrust .. which will cause some difficulty with roll control. Just what you can or can't do, roll-wise, will be Type specific
The 5° limit is a certification consideration. If you happen to be back in real Vmc territory, anticipate that you will need this bank to make things work .. otherwise the real Vmc will vary. Vmc is about not losing control .. climb capability is quite secondary. Once you get back to a speed where climb might be feasible, a bank angle of around 2-3° generally results in the best compromise for climb performance.
I think you mean Vmca
Same deal .. evolving acronyms
Thus the Vmca issue is alleviated by the turn into the good engine, and that meshes with the old "dead engine" rule
Not quite. Vmc is still waiting in the wings to bite you on the tail. However, it is reasonable to opine that things might get away from the pilot more readily with a turn towards the dead engine in the event of mishandling etc. It follows that risk management considerations might dictate the preference to turn to the live engine.