diff --git a/.cirrus.yml b/.cirrus.yml
index 94d1811baef64..9ff908ffe870e 100644
--- a/.cirrus.yml
+++ b/.cirrus.yml
@@ -263,6 +263,42 @@ task:
     - ./ci/cirrus.sh
   << : *CAT_LOGS
 
+task:
+  << : *LINUX_CONTAINER
+  env:
+    WRAPPER_CMD: wine
+    WERROR_CFLAGS: -WX
+    WITH_VALGRIND: no
+    ECDH: yes
+    RECOVERY: yes
+    EXPERIMENTAL: yes
+    SCHNORRSIG: yes
+    CTIMETEST: no
+    # Set non-essential options that affect the CLI messages here.
+    # (They depend on the user's taste, so we don't want to set them automatically in configure.ac.)
+    CFLAGS: -nologo -diagnostics:caret
+    LDFLAGS: -XCClinker -nologo -XCClinker -diagnostics:caret
+  # Use a MinGW-w64 host to tell ./configure we're building for Windows.
+  # This will detect some MinGW-w64 tools but then make will need only
+  # the MSVC tools CC, AR and NM as specified below.
+  matrix:
+    - name: "x86_64 (MSVC): Windows (Debian stable, Wine)"
+      env:
+        HOST: x86_64-w64-mingw32
+        CC: /opt/msvc/bin/x64/cl
+        AR: /opt/msvc/bin/x64/lib
+        NM: /opt/msvc/bin/x64/dumpbin -symbols -headers
+    - name: "i686 (MSVC): Windows (Debian stable, Wine)"
+      env:
+        HOST: i686-w64-mingw32
+        CC: /opt/msvc/bin/x86/cl
+        AR: /opt/msvc/bin/x86/lib
+        NM: /opt/msvc/bin/x86/dumpbin -symbols -headers
+  << : *MERGE_BASE
+  test_script:
+    - ./ci/cirrus.sh
+  << : *CAT_LOGS
+
 # Sanitizers
 task:
   << : *LINUX_CONTAINER
diff --git a/ci/cirrus.sh b/ci/cirrus.sh
index b85f012d3f3c0..929c16d511d51 100755
--- a/ci/cirrus.sh
+++ b/ci/cirrus.sh
@@ -9,6 +9,7 @@ env >> test_env.log
 
 $CC -v || true
 valgrind --version || true
+$WRAPPER_CMD --version || true
 
 ./autogen.sh
 
diff --git a/ci/linux-debian.Dockerfile b/ci/linux-debian.Dockerfile
index 5cccbb5565f03..a83a4e36db040 100644
--- a/ci/linux-debian.Dockerfile
+++ b/ci/linux-debian.Dockerfile
@@ -1,15 +1,14 @@
 FROM debian:stable
 
-RUN dpkg --add-architecture i386
-RUN dpkg --add-architecture s390x
-RUN dpkg --add-architecture armhf
-RUN dpkg --add-architecture arm64
-RUN dpkg --add-architecture ppc64el
-RUN apt-get update
+RUN dpkg --add-architecture i386 && \
+    dpkg --add-architecture s390x && \
+    dpkg --add-architecture armhf && \
+    dpkg --add-architecture arm64 && \
+    dpkg --add-architecture ppc64el
 
 # dkpg-dev: to make pkg-config work in cross-builds
 # llvm: for llvm-symbolizer, which is used by clang's UBSan for symbolized stack traces
-RUN apt-get install --no-install-recommends --no-upgrade -y \
+RUN apt-get update && apt-get install --no-install-recommends -y \
         git ca-certificates \
         make automake libtool pkg-config dpkg-dev valgrind qemu-user \
         gcc clang llvm libc6-dbg \
@@ -19,8 +18,20 @@ RUN apt-get install --no-install-recommends --no-upgrade -y \
         gcc-arm-linux-gnueabihf libc6-dev-armhf-cross libc6-dbg:armhf \
         gcc-aarch64-linux-gnu libc6-dev-arm64-cross libc6-dbg:arm64 \
         gcc-powerpc64le-linux-gnu libc6-dev-ppc64el-cross libc6-dbg:ppc64el \
-        wine gcc-mingw-w64-x86-64 \
+        gcc-mingw-w64-x86-64-win32 wine64 wine \
+        gcc-mingw-w64-i686-win32 wine32 \
         sagemath
 
-# Run a dummy command in wine to make it set up configuration
-RUN wine64-stable xcopy || true
+WORKDIR /root
+# The "wine" package provides a convience wrapper that we need
+RUN apt-get update && apt-get install --no-install-recommends -y \
+        git ca-certificates wine64 wine python3-simplejson python3-six msitools winbind procps && \
+    git clone https://github.com/mstorsjo/msvc-wine && \
+    mkdir /opt/msvc && \
+    python3 msvc-wine/vsdownload.py --accept-license --dest /opt/msvc Microsoft.VisualStudio.Workload.VCTools && \
+    msvc-wine/install.sh /opt/msvc
+
+# Initialize the wine environment. Wait until the wineserver process has
+# exited before closing the session, to avoid corrupting the wine prefix.
+RUN wine64 wineboot --init && \
+    while (ps -A | grep wineserver) > /dev/null; do sleep 1; done