Open Gm tuning
The open Gm tuning also known as Cross-note G tuning is not as common as the Open G tuning.
To get the open G minor tuning on your guitar you tune it like this: DGDGBbD. The strings that differ from the standard should be turned so the string tension is decreased and not the opposite.
More info
Including an octave perspective, the tuning is written D2-G2-D3-G3-Bb3-D4, meaning that the lowest string is a D note on the second octave, the second lowest string is a G note on the second octave and so on. Standard tuning reference: E2-A2-D3-G3-B3-E4.
Chords
Gm
Gm6
Gm7
Gm9
Bbmaj7
Cm
Cm/D
Dm
Ebmaj7
Eb9
Fadd9
F9
Comments
The Cm chord is more correctly written as Cm/G. By also include the open low e-string it would be a Cm/Eb.
The Cm chords is more correctly written as Dmadd11. A regular Dm chord could be played as xx0243 or 0x0243.
The Ebmaj7 chord could also be played as 100000 or 100001. The Eb9 chord is more correctly written as Eb9/D. Ebmaj9/D could be played as 000343.
The Fadd9 and F9 chords are more correctly written as Fadd9/D and F9/D. F7/D can be played as 021223 or 020553.
Other chord possibilities in Open G minor
tuning:
Gm/A: X20000
Gm9/D: 000203
Gmadd2/D: X77007
GmMaj11/D: 054000
Gsus2/Eb: X87050
Bb6: X33000
Bbmaj7/D: XX0203
Csus2/D: 050040
C7: X55005
Chord progressions
Two chord progressions in this tuning:
Gm9 – Ebmaj7 – Bbmaj7 – Gm7
Gm7 – Dm – Cm/D – Bbmaj7
For more progressions, see The Alternate Tunings ebook.
Open Am tuning
Another option is the Open Am tuning, which you can play in by putting a capo on the second fret when your guitar is tuned to Open Gm. Otherwise, the tuning for Open A minor is this: EAEACE. It can also be an advantage to use a capo compared to tuning in this case since two of the strings will increase the tension, which can result in breakage of the strings.
See also Open Dm tuning