iOS 13 Syncs Calendar and Contacts but not Reminders

Ok, we found the solution (many thanks to @Bernie_O ): the post by “fixitnowyes” on Apple-discussions gives the necessary instructions to create a certificate which will be accepted by my iOS 13 devices! I am quite happy to have syncing reminders on my local server again.
Don’t foget to import the new certificate to the iOS device and switch on the “trust” (settings / general / about / scroll down to “certificate trust settings”).

Edit: This is the content of the above-mentioned post or - if german-speaking - read a Blog-Artikel darüber :

Create a file “server-selfsigned-CA.cnf” with content (replace your.servers.domain.com with domain name of server and 10.0.0.1 with IP of your server):

[ req ]
default_bits        = 4096
default_keyfile     = server-selfsigned-CA.key
default_md          = sha256
default_days        = 824
encrypt_key         = no
distinguished_name  = subject
req_extensions      = req_ext
x509_extensions     = x509_ext
string_mask         = utf8only
prompt              = no

# The Subject DN can be formed using X501 or RFC 4514 (see RFC 4519 for a description).
#   Its sort of a mashup. For example, RFC 4514 does not provide emailAddress.
[ subject ]
OU                     = Company Inc.
countryName            = DE
stateOrProvinceName    = Berlin
localityName           = Berlin
organizationName       = Company Inc.

# Use a friendly name here because it's presented to the user. The server's DNS
#   names are placed in Subject Alternate Names. Plus, DNS names here is deprecated
#   by both IETF and CA/Browser Forums. If you place a DNS name here, then you
#   must include the DNS name in the SAN too (otherwise, Chrome and others that
#   strictly follow the CA/Browser Baseline Requirements will fail).
commonName              = aFriendlyName
emailAddress            = some@mail.com


# Section x509_ext is used when generating a self-signed certificate. I.e., openssl req -x509 ...
[ x509_ext ]
subjectKeyIdentifier      = hash
#authorityKeyIdentifier    = keyid,issuer
authorityKeyIdentifier    = keyid:always,issuer


# You only need digitalSignature below. *If* you don't allow
#   RSA Key transport (i.e., you use ephemeral cipher suites), then
#   omit keyEncipherment because that's key transport.
basicConstraints    = critical, CA:TRUE
keyUsage            = critical, digitalSignature, keyEncipherment, cRLSign, keyCertSign
subjectAltName      = DNS:your.servers.domain.com, DNS:10.0.0.1
# nsComment         = "OpenSSL Generated Certificate"
extendedKeyUsage    = serverAuth


# RFC 5280, Section 4.2.1.12 makes EKU optional
#   CA/Browser Baseline Requirements, Appendix (B)(3)(G) makes me confused
#   In either case, you probably only need serverAuth.
# extendedKeyUsage    = serverAuth, clientAuth
extendedKeyUsage    = TLS Web Server Authentication


# Section req_ext is used when generating a certificate signing request. I.e., openssl req ...
[ req_ext ]
subjectKeyIdentifier  = hash
basicConstraints      = CA:FALSE
keyUsage              = digitalSignature, keyEncipherment
subjectAltName        = DNS:your.servers.domain.com, DNS:10.0.0.1
nsComment             = "OpenSSL Generated Certificate"


# RFC 5280, Section 4.2.1.12 makes EKU optional
#   CA/Browser Baseline Requirements, Appendix (B)(3)(G) makes me confused
#   In either case, you probably only need serverAuth.
#extendedKeyUsage    = serverAuth, clientAuth


# [ alternate_names ]
# DNS.1       = example.com
# DNS.2       = www.example.com
# DNS.3       = mail.example.com
# DNS.4       = ftp.example.com


# Add these if you need them. But usually you don't want them or
#   need them in production. You may need them for development.
# DNS.5       = localhost
# DNS.6       = localhost.localdomain
# DNS.7       = 127.0.0.1

Generate a new certificate using the .cnf file:

sudo openssl req -config server-selfsigned-CA.cnf -new -x509 -out server-selfsigned-CA.crt -days 825

You can check if the certificate was created correctly with:

openssl x509 -in server-selfsigned-CA.crt -text -noout

Import the file “server-selfsigned-CA.crt” to iOS device (I used a SMB connection and the new “files”-app).

On the iOS device, go to settings and install the certificate.

Then, go to settings / general / about / scroll down to “certificate trust settings” and switch on trust for the certificate.

And that’s it, reminders app will sync again.