Here’s Creifelds for a definition (the relevant sections of the StPO are given in English in the comment on February 12):

bq. Die Strafverfolgungsbehörden (StA, Polizei, Finanzamt usw.) haben nach §§ 152 II, 160, 163 StPO, § 386 AO bei Verdacht einer Straftat von Amts wegen, also auch ohne Anzeige, einzuschreiten. Dazu ist die StA befugt, von allen Behörden Auskunft zu verlangen und Ermittlungen jeder Art vorzunehmen. Ausnahmen gelten nach dem ®Opportunitätsprinzip insbes. in ®Bagatellstrafsachen. Das L. soll die Einhaltung des Grundsatzes der Gleichheit vor dem Gesetz (Art. 3 I GG) sichern. Es ist insofern die notwendige Ergänzung zum Anklagemonopol der StA, die im Einzelfall nach Abschluss des ®Ermittlungsverfahrens darüber entscheidet, ob das Strafverfahren durchgeführt werden soll …

Suggestions:
legality principle (this is definitely used sometimes but may be confusing)
principle of mandatory prosecution (too much emphasis on prosecution)
principle of prosecution ex officio
principle of mandatory investigation (too much emphasis on investigation? The duty goes further than investigating)

principle of discretionary prosecution
principle of prosecutorial discretion (I didn’t see the point of this when I first heard it, but prosecutorial avoids that problem of emphasis on the prosecution itself)

The terms used in practice are principle of mandatory / compulsory prosecution / legality and principle of expediency / opportunity principle. Here is a book on the criminal process by Ashworth and Redmayne, ina chapter on gatekeeping and diversion:

bq. In many European systems of law there is a doctrinal contrast between the principle of compulsory prosecution (sometimes called the principle of legality) and the principle of expediency (sometimes called the opportunity principle). Many systems, such as the German and the Austrian, have placed great emphasis on the principle of compulsory prosecution. …In theory all those who commit offences are brought before the courts for an open determination of guilt and (if convicted) for sentencing, and there is no broad discretionary power to avoid prosecution – on the grounds that this might lead to local variations …

Here is a nice online comparison of the German and U.S. criminal justice systems (PDF), published by the U.S. Department of Justice:

bq. English language works discussing German prosecutors generally focus on he German concept of obligatory prosecution (“legality principle”), a principle that contrasts sharply with the American emphasis on prosecutorial discretion (described by the Germans as the “opportunity principle”).