Foreign investment in asset management is allowed subject to local incorporation. Managers and trustees of 'internal' (domestic) collective investment must be locally incorporated.
"5. Asset managers to be registered
(1) Subject to this section, no person shall carry on the business of asset management unless the person–
(a) is incorporated as a private or public company in terms of the Companies Act [Chapter 24:03]; and
(b) in the case of–
(i) a person who carried on the business of asset management before the 17th September, 2004, is registered in terms of Statutory Instrument 16 of 2004 on or before the 31st January, 2004;
(ii) a person who wishes to carry on the business of asset management after the 17th September, 2004, is registered in terms of this Act before commencing the business.[...]" (Asset Management Act (Chapter 24:26) of 2004, as amended)
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"Part IV – Internal schemes
10. Application of Part IV
This Part shall apply to internal schemes, that is to say, to any unit trust scheme which is constituted by a trust deed executed in Zimbabwe and which is operated wholly or mainly in Zimbabwe.
[...]
12. Manager and trustee
(1) No person other than a company which is— (a) registered and resident in Zimbabwe; and (b) licensed in terms of section thirteen; shall be or act as the manager or trustee of an internal scheme.
[...]
13. Licensing of manager and trustee [...]
(2) Upon receipt of an application in terms of subsection (1), the Chief Executive Officer shall issue a licence to the applicant permitting it to be the manager or trustee, as the case may be, of an internal scheme if the Chief Executive Officer is satisfied—
(a) that the applicant is a company registered in Zimbabwe and that its affairs are managed and controlled in Zimbabwe; and
[...]
Part V – External schemes
19. Application of Part V
This Part shall apply to external schemes, that is to say, to any unit trust scheme which is established and managed outside Zimbabwe. 
[...]
21. Manager of external scheme to maintain local office
The manager of every registered external scheme shall maintain an office in Zimbabwe where—
(a) the scheme’s operations in Zimbabwe shall be conducted; and
(b) all communications and notices may be addressed and all process may be served; and
shall inform the Chief Executive Officer of the address of the office and of any change in that address.[...]" (Collective Investment Schemes Act (Chapter 24:19) of 1998, as amended)

Asset Management Act (Chapter 24:26) of 2004, as amended
Collective Investment Schemes Act (Chapter 24:19) of 1998, as amended
[<a href="https://zimlii.org/akn/zw/act/2004/15/eng@2016-12-31">Act</a>],[<a href="https://zimlii.org/akn/zw/act/1997/25/eng@2016-12-31">Act</a>]