What is Minimum Control speed V_MC, or V_MCA?

Bank angle versus sideslip and V_MCA for a sample airplane when engine #1 is inoperative.
This graph shows:
When the Indicated Airspeed is near AFM published V_MCA, then bank 4° away from the inoperative engine to avoid loss of control and for minimum drag, i.e. for max. Rate of Climb.

V_MCA is a constant number, but only in the AFM. In flight, the actual V_MCA varies with bank angle, engine thrust, rudder deflection, and other variables. This graph shows (actual) V_MCA and sideslip angle, both versus bank angle of a sample airplane after failure of the left engine (#1) while the asymmetrical thrust and rudder are maximal.
The airspeed that results from the bank angle for which the sideslip is zero (minimum drag, max. ROC) is the V_MCA that will be published in the AFM (85 kt). At bank angles larger than 6° away from the failed engine, for this sample airplane, the sideslip angle increases to 14°, being the fin stall angle of attack (with deflected rudder). The airspeed needs to be increased to prevent the fin from stalling at larger bank angles - the actual V_MCA increases (blue line).
Notice that the actual V_MCA for wings level (100 kt) of this sample airplane is 15 kt higher than the AFM published V_MCA.
This leads to the above announced condition that the pilot should not maintain a bank angle of maximum 5° (either side) as is presented in V_MC definitions in most AFM's, but the exact bank angle that was used to design the vertical tail and at which the drag is minimal (in this example 4°, usually 5°), when the airspeed decreases to or is V_MCA while the asymmetrical thrust is maximal. At higher airspeeds, the bank angle can be smaller; at V_YSE mostly 3°.
This small bank angle does not result in a turn, but reduces both the drag and V_MCA.

FAR and EASA/CS 23.149 and equivalent present the definition of V_MCA for the design and certification of multi-engine airplanes that is also inappropriately copied into most AFM's:

V_MC is the calibrated airspeed at which, when the critical engine is suddenly made inoperative, it is possible to maintain control of the airplane with that engine still inoperative, and thereafter maintain straight flight at the same speed with an angle of bank of not more than 5 degrees.

Once the airplane is designed and build, the selected tail size imposes a limitation on, i.e. a constraint to, pilots.  The V_MCA definition for use by pilots is therefore different than the V_MCA definition out of FAR/CS 23.149 that is for manufacturers, for designing and certification of airplanes.

V_MCA is the minimum speed for maintaining straight flight when an engine fails or is inoperative and the corresponding opposite engine is set to provide maximum thrust, provided a bank angle is being maintained of 3 – 5 degrees (exact number to be provided by the manufacturer) away from the inoperative engine.

In addition, the manufacturer should specify the configuration for which V_MCA and other published V_MC's are valid.

On the airspeed indicator of Part 23 twin-engine airplanes, the standardized AFM-published V_MCA is indicated by a red radial line, in this example at 80 kt. However, neither a placard on the instrument panel nor a note or warning in the AFM tells the pilot that the redlined V_MCA is valid only if a bank angle of 3 to 5 degrees (to be specified by the manufacturer) is maintained away from the inoperative engine. A larger bank angle, or a bank angle into the inoperative engine results in a much higher actual V_MCA, a 14 or more degree sideslip, hence large drag, and to the loss of climb performance and possibly loss of control after which an accident cannot be avoided (when asymmetrical thrust is not reduced).

The airspeed for maximum single-engine rate of climb V_YSE is indicated by a blue radial line, here at 105 kt. In the legend of some Performance Data Tables of Graphs, a note tells the pilot that the presented performance data, including the performance at V_YSE, are valid only if a small bank angle is being maintained of 2 - 3 degrees away from the inoperative engine. For other bank angles, the maximum climb performance or the performance to maintain altitude (i.e. to prevent drifting down) is not guaranteed.

Dr. Jan Roskam (KU): "The V_MCA value ultimately used ties take-off performance to engine-out controllability."

If the pointer is at or near the red line and the thrust on the remaining engine(s) is or is increased to maximum, only straight flight should be maintained while maintaining a bank angle of 3 to 5 degrees away from the inoperative engine, depending on airplane type and airspeed V_YSE and V_MCA resp.).
For turning safely while the asymmetrical thrust is high, gain altitude first during straight flight to allow for some altitude loss during reduced thrust turns, because of the increased sideslip (drag) during turns. It is safer to reduce the thrust a little during the turns to keep the actual V_MCA low. Also consider a long straight-in approach rather than a tight final turn during which the thrust might have to be increased to maximum for maintaining the glide path (and control will be lost because actual V_MCA increases above the indicated airspeed). This happened many times.

This note is included in the legend of the Climb Performance Chart - One Engine Operating in the Piper PA-44 Pilot's Information Manual. It is included, because not maintaining this bank angle renders the presented performance data invalid; the airplane might not even be able to maintain altitude. The bank angle is smaller than 5 degrees, because the presented performance data requires V_YSE, the blue line speed, which is higher than V_MCA. The vertical tail is more effective at higher speed.
Keeping the wings level or turning means loss of performance; altitude cannot be maintained on most multi-engine airplanes if this NOTE is neglected. The reason why this NOTE is included is explained in the papers presented on the Downloads page.

A similar placard is to be installed in full view of pilots of Part 23 commuter airplanes to comply with Aviation Regulations (23.1563). The required small bank angle for the listed V_MCA to be valid is regrettably not included on the placard, because this is not required by Aviation Regulations, but is essential for flight safety and performance.

Not maintaining the small bank angle (i.e. straight flight) at airspeeds as low as V_MCA, while the power setting of the remaining engine is high, is the real cause of most engine failure related accidents.

It is recommended to require a placard like this one in all Part 23 airplanes.

More options are possible.